Lecture Notes on Quantum Mechanics: Wave Reflection, Virtual Lab, Linear Algebra and Hilbert Spaces

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February 5, 2025

Introduction

This lecture covers fundamental concepts essential for understanding quantum mechanics. We begin by examining wave reflection and transmission at an interface, focusing on energy conservation and the phase difference between transmitted and reflected waves. We then explore a virtual quantum laboratory demonstration to visualize quantum phenomena such as interference, non-destructive measurement, and the Deutsch-Jozsa algorithm. The tri-polarized lens paradox will be discussed to highlight the distinctions between classical and quantum logic. Following this, we will recap linear algebra, introducing vector spaces, Dirac notation, inner products, linear combinations, and bases. Finally, we will define Hilbert spaces and linear operators, including projector operators and the completeness relation. This lecture aims to provide a comprehensive overview, bridging wave phenomena with the abstract mathematical framework of quantum mechanics.

Wave Reflection and Transmission

Reflection and Transmission at an Interface

When a ray of light encounters an interface between two media, it is partially reflected and partially transmitted. Consider an incident wave described by the wave function $\Psi_0(x,t) = \cos(\omega t - kx)$. Upon interaction with the interface, this wave gives rise to a reflected wave $\Psi_R(x,t)$ and a transmitted wave $\Psi_T(x,t)$, both oscillating at the same frequency $\omega$. Changes in intensity and phase may occur for both the reflected and transmitted waves.

Reflection and Transmission at an Interface

Energy Conservation and Intensity

Wave intensity is proportional to the square of the wave function amplitude and represents energy flux. Assuming energy conservation at the interface, the sum of the reflected and transmitted intensities must equal the incident intensity. Let $I_0$, $I_R$, and $I_T$ denote the intensities of the incident, reflected, and transmitted waves, respectively. Energy conservation dictates: \[I_R + I_T = I_0\] Since intensity is proportional to $|\Psi|^2$, we can express this in terms of wave function amplitudes $\Psi_0$, $\Psi_R$, and $\Psi_T$: \[|\Psi_R|^2 + |\Psi_T|^2 = |\Psi_0|^2\] Normalizing by $|\Psi_0|^2$, and defining amplitude reflection coefficient $r = \frac{|\Psi_R|}{|\Psi_0|}$ and amplitude transmission coefficient $t = \frac{|\Psi_T|}{|\Psi_0|}$, we obtain: \[r^2 + t^2 = 1\] This relationship expresses energy conservation in terms of amplitude coefficients.

Phase Difference from Symmetric Two-Ray System

To illustrate the phase difference between transmitted and reflected waves, consider a symmetric system with two identical incoming rays, as proposed by Didin Chain or De George.

Symmetric Setup and Superposition

Imagine two identical incoming rays with the same intensity and phase incident on a symmetric beam splitter. Each ray is partially reflected and transmitted. Due to the symmetry and superposition principle, the output wave function amplitude $\Psi_{out}$ is proportional to $(T+R)\Psi_0$, where $\Psi_0$ is the incoming wave amplitude, and $T$ and $R$ are the transmission and reflection coefficients respectively. The output intensity $I_{out} \propto |\Psi_{out}|^2 = |\Psi_0|^2 |T+R|^2$.

Symmetric Two-Ray System

Phase Difference Derivation

Assuming a 50/50 beam splitter and equating output intensity considerations with energy conservation, we can impose the condition $|T+R|^2 = |T|^2 + |R|^2$. Expanding the left side: \[\begin{aligned}|T+R|^2 &= (T+R)(T^*+R^*) \\&= TT^* + TR^* + RT^* + RR^* \\&= |T|^2 + |R|^2 + TR^* + RT^*\end{aligned}\] For this to equal $|T|^2 + |R|^2$, we require $TR^* + RT^* = 0$, which simplifies to $2 \Re(TR^*) = 0$. Let $T = |T|e^{i\phi_T}$ and $R = |R|e^{i\phi_R}$. Then $TR^* = |T||R|e^{i(\phi_T - \phi_R)}$, and: \[2 \Re(|T||R|e^{i(\phi_T - \phi_R)}) = 2|T||R|\cos(\phi_T - \phi_R) = 0\] This implies $\cos(\phi_T - \phi_R) = 0$, meaning the phase difference $\phi_T - \phi_R$ is an odd multiple of $\pi/2$. The simplest non-trivial case is $\phi_T - \phi_R = \pm \frac{\pi}{2}$.

Phase Shift and Imaginary Unit

Choosing $\phi_T - \phi_R = \frac{\pi}{2}$, we have $e^{i(\phi_T - \phi_R)} = e^{i\pi/2} = i$. This indicates a $\pi/2$ phase difference between transmitted and reflected waves. We can represent this by setting $R = i|R|$ and $T = |T|$ (or vice versa). This phase shift is fundamentally linked to the imaginary unit $i$, which is intrinsic to the time-dependent Schrödinger equation and wave phenomena in quantum mechanics. Even in simplified models, neglecting complex boundary conditions and Maxwell’s equations, energy conservation arguments lead to this phase difference, highlighting the fundamental role of the imaginary unit in describing wave behavior.

Virtual Quantum Laboratory Demonstration

Introduction to the Virtual Lab Environment

A virtual quantum laboratory, accessible online, offers an interactive platform for simulating and visualizing quantum experiments. This environment allows users to manipulate quantum optical elements such as mirrors, beam splitters, and polarizers to explore phenomena like interference, non-destructive measurement, and quantum algorithms. The user interface features a selection of quantum components, sources, and detectors, enabling the setup and execution of experiments with real-time probability and outcome displays after repeated runs. Users can typically search for this "virtual laboratory" online to access the platform and begin experimenting.

Interference Demonstration

One of the primary demonstrations within the virtual lab is interference. By configuring an interferometer, such as a Mach-Zehnder interferometer, the wave nature of photons becomes directly observable. A typical setup involves a photon source directed into a beam splitter, which separates the photon’s path into two arms. These paths are subsequently recombined at another beam splitter, with detectors positioned at the output ports to measure photon detection probabilities.

By precisely adjusting the path lengths, constructive interference can be achieved at one detector, resulting in a 100% photon detection rate at that port, while the other detector registers no detections due to destructive interference. This vividly demonstrates the principles of superposition and the wave-like behavior of quantum particles.

Non-Destructive Measurement

The virtual lab also facilitates demonstrations of non-destructive measurement. By employing a specialized detector that interacts weakly with a photon, it is possible to gain path information without absorbing or destroying the photon. Observing the photon in one path in such a setup can demonstrably alter the interference pattern at the output detectors. This highlights a key distinction from classical physics: quantum measurement is not a passive observation but an interaction that can fundamentally change the state and behavior of the quantum system.

Deutsch-Jozsa Algorithm Simulation

The virtual lab provides a simulation of the Deutsch-Jozsa algorithm, a foundational quantum algorithm. In this virtual implementation, quantum gates are represented by optical elements. Specifically, the oracle in the Deutsch-Jozsa algorithm is represented by two distinct optical objects within the lab. While not implemented with standard Hadamard gates directly, components resembling square root of NOT gates are used in conjunction with these oracle objects.

The Deutsch-Jozsa algorithm in this setting aims to determine whether a function is constant or balanced. The setup visualizes the principle of querying the oracle in superposition, effectively evaluating the function for both inputs (0 and 1) simultaneously within a single quantum query. The experimental configuration typically includes elements analogous to Hadamard gates before and after the oracle components, culminating in a measurement to determine the algorithm’s output. This simulation allows users to observe how superposition and interference are harnessed to achieve a quantum speedup compared to classical approaches for this problem.

Tri-polarized Lens Paradox

The tri-polarized lens paradox illustrates a non-classical outcome of quantum measurement that contrasts sharply with classical intuition.

Classical vs. Quantum Predictions

Consider three polarizing lenses: A (vertically polarized), B (horizontally polarized), and C (polarized at $45^\circ$ to the vertical). Classically, if light passes through lens A and then lens B, no light is transmitted because a horizontal polarizer blocks vertically polarized light. If we represent vertically polarized photons as set A and horizontally polarized photons as set B, their intersection is empty ($A \cap B = \emptyset$). Classical logic dictates that inserting an intermediate filter C should not allow transmission when $A \cap B$ is already empty; thus, $A \cap C \cap B$ should also be empty.

Quantum Measurement and State Modification

Quantum mechanically, however, placing lens C between lenses A and B allows some light to pass through all three. After passing lens A, photons are vertically polarized. Upon encountering lens C ($45^\circ$), the vertical polarization state is projected onto the $45^\circ$ polarization direction. There is a non-zero probability for photons to pass lens C and become $45^\circ$ polarized. Subsequently, when these $45^\circ$ polarized photons reach lens B (horizontal), there is again a non-zero probability of transmission, resulting in horizontally polarized photons.

Therefore, inserting lens C creates a transmission path through all three lenses, even though lenses A and B alone would block all light. This paradox arises because quantum measurement is not a passive selection but an active state modification. Measurement by lens C projects the photon into a $45^\circ$ polarized state, which has a non-zero component in the horizontal polarization direction, enabling transmission through lens B. This demonstrates that quantum measurements fundamentally alter quantum states, and classical logic does not universally apply to quantum phenomena.

Linear Algebra Recap

Vector Spaces and Dirac Notation

In quantum mechanics, physical states are represented as vectors within a vector space. Dirac notation, also known as bra-ket notation, is the standard convention for representing these vectors.

Ket and Bra Vectors

A vector in a complex vector space is denoted by a ket, symbolized as $\left| \psi \right\rangle$ (e.g., $\left| v \right\rangle$). In component form, a ket is represented as a column vector. The dual vector, known as a bra, is denoted as $\left\langle \psi \right|$ and is the conjugate transpose (Hermitian conjugate) of the ket vector. In component form, a bra is represented as a row vector with complex conjugated entries.

For a ket vector $\left| v \right\rangle = \begin{pmatrix} c_1 \\ c_2 \\ \vdots \\ c_n \end{pmatrix}$, its corresponding bra vector is $\left\langle v \right| = \begin{pmatrix} c_1^* & c_2^* & \cdots & c_n^* \end{pmatrix}$, where $c_i \in \mathbb{C}$ and $c_i^*$ is the complex conjugate of $c_i$.

Vector Operations

Vector spaces are closed under vector addition and scalar multiplication, fundamental operations in linear algebra.

Vector Addition: For any two vectors $\left| u \right\rangle, \left| v \right\rangle$ in a vector space $V$, their sum $\left| u \right\rangle + \left| v \right\rangle$ is also in $V$. Vector addition is performed component-wise.
Scalar Multiplication: For a vector $\left| v \right\rangle \in V$ and a scalar $c \in \mathbb{C}$, the scalar product $c\left| v \right\rangle$ is also in $V$. Scalar multiplication involves multiplying each component of $\left| v \right\rangle$ by $c$.

Inner Product and Orthogonality

The inner product, or scalar product, generalizes the dot product to complex vector spaces. It maps two vectors to a scalar and defines orthogonality.

Definition of Inner Product

Inner Product: The inner product of two vectors $\left| x \right\rangle$ and $\left| y \right\rangle$ in a complex vector space, denoted as $\left\langle x \middle| y \right\rangle$, is defined as: \[\left\langle x \middle| y \right\rangle = \left\langle x \right| \left| y \right\rangle = \sum_{i} x_i^* y_i\] where $\left| x \right\rangle = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}$ and $\left| y \right\rangle = \begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix}$.

Properties of the Inner Product

The inner product satisfies the following properties for vectors $\left| x \right\rangle, \left| y \right\rangle, \left| z \right\rangle$ and scalars $c_1, c_2 \in \mathbb{C}$:

Conjugate Symmetry: $\left\langle x \middle| y \right\rangle = \left\langle y \middle| x \right\rangle^*$
Linearity in the second argument: $\left\langle x \middle| c_1\left| y_1 \right\rangle + c_2\left| y_2 \right\rangle \right\rangle = c_1\left\langle x \middle| y_1 \right\rangle + c_2\left\langle x \middle| y_2 \right\rangle$
Conjugate Linearity in the first argument: $\left\langle c_1\left| x_1 \right\rangle + c_2\left| x_2 \right\rangle \middle| y \right\rangle = c_1^*\left\langle x_1 \middle| y \right\rangle + c_2^*\left\langle x_2 \middle| y \right\rangle$
Positive-Definiteness: $\left\langle x \middle| x \right\rangle \geq 0$, with $\left\langle x \middle| x \right\rangle = 0$ if and only if $\left| x \right\rangle = \left| 0 \right\rangle$.

Note the non-commutativity: $\left\langle x \middle| y \right\rangle \neq \left\langle y \middle| x \right\rangle$ in general, but rather $\left\langle x \middle| y \right\rangle = \left\langle y \middle| x \right\rangle^*$.

Orthogonality

Two vectors $\left| x \right\rangle$ and $\left| y \right\rangle$ are orthogonal if their inner product is zero: \[\left\langle x \middle| y \right\rangle = 0\]

Linear Combinations and Linear Independence

Linear combinations and linear independence are essential for characterizing vector space structure.

Linear Combinations and Linear Manifolds (Subspaces)

A linear combination of vectors $\{\left| v_1 \right\rangle, \left| v_2 \right\rangle, \ldots, \left| v_n \right\rangle\}$ is a vector $\left| v \right\rangle$ expressed as: \[\left| v \right\rangle = c_1\left| v_1 \right\rangle + c_2\left| v_2 \right\rangle + \cdots + c_n\left| v_n \right\rangle = \sum_{i=1}^{n} c_i\left| v_i \right\rangle\] where $c_i \in \mathbb{C}$ are scalar coefficients.

The set of all possible linear combinations of $\{\left| v_1 \right\rangle, \left| v_2 \right\rangle, \ldots, \left| v_n \right\rangle\}$ forms a linear manifold, or subspace, spanned by these vectors.

Linear Dependence and Independence

A set of vectors $\{\left| v_1 \right\rangle, \left| v_2 \right\rangle, \ldots, \left| v_n \right\rangle\}$ is linearly dependent if there exist coefficients $\{c_1, c_2, \ldots, c_n\}$, not all zero, such that: \[\sum_{i=1}^{n} c_i\left| v_i \right\rangle = \left| 0 \right\rangle\] If the only solution is $c_1 = c_2 = \cdots = c_n = 0$, the set is linearly independent. Equivalently, linear dependence means at least one vector can be expressed as a linear combination of the others; otherwise, they are linearly independent.

Checking Linear Independence

Linear independence can be checked using methods like Gaussian elimination. Consider vectors $\left| v_1 \right\rangle = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix}$, $\left| v_2 \right\rangle = \begin{pmatrix} 4 \\ 7 \\ 4 \end{pmatrix}$, $\left| v_3 \right\rangle = \begin{pmatrix} 2 \\ 5 \\ 3 \end{pmatrix}$. We transform them using linear combinations:

Keep $\left| v_1 \right\rangle$ as is: $\left| v_1 \right\rangle = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix}$.
Replace $\left| v_2 \right\rangle$ with $\left| v_2' \right\rangle = \left| v_2 \right\rangle - 4\left| v_1 \right\rangle = \begin{pmatrix} 0 \\ -5 \\ -4 \end{pmatrix}$.
Replace $\left| v_3 \right\rangle$ with $\left| v_3' \right\rangle = \left| v_3 \right\rangle - 2\left| v_1 \right\rangle = \begin{pmatrix} 0 \\ -1 \\ -1 \end{pmatrix}$.
Replace $\left| v_3' \right\rangle$ with $\left| v_3'' \right\rangle = \left| v_3' \right\rangle - \frac{1}{5}\left| v_2' \right\rangle = \begin{pmatrix} 0 \\ 0 \\ -1/5 \end{pmatrix}$.

The transformed set is $\left| v_1 \right\rangle = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix}$, $\left| v_2' \right\rangle = \begin{pmatrix} 0 \\ -5 \\ -4 \end{pmatrix}$, $\left| v_3'' \right\rangle = \begin{pmatrix} 0 \\ 0 \\ -1/5 \end{pmatrix}$. Since we have non-zero components along different dimensions after transformation, the original vectors are linearly independent. If a zero vector were obtained during this process, it would indicate linear dependence.

Basis and Orthonormal Basis

A basis is a minimal set of vectors that spans the entire vector space. An orthonormal basis is a basis composed of orthogonal and normalized vectors, which simplifies many calculations.

Definition of a Basis

Basis: A set of vectors $\{\left| b_1 \right\rangle, \left| b_2 \right\rangle, \ldots, \left| b_n \right\rangle\}$ is a basis for a vector space $V$ if it satisfies two conditions:

Linear Independence: $\{\left| b_1 \right\rangle, \left| b_2 \right\rangle, \ldots, \left| b_n \right\rangle\}$ is a linearly independent set.
Spanning: $\{\left| b_1 \right\rangle, \left| b_2 \right\rangle, \ldots, \left| b_n \right\rangle\}$ spans $V$, meaning any vector $\left| v \right\rangle \in V$ can be expressed as a linear combination of $\{\left| b_1 \right\rangle, \left| b_2 \right\rangle, \ldots, \left| b_n \right\rangle\}$.

In an $n$-dimensional vector space, any set of $n$ linearly independent vectors constitutes a basis.

Orthonormal Basis and Kronecker Delta

A basis $\{\left| u_1 \right\rangle, \left| u_2 \right\rangle, \ldots, \left| u_n \right\rangle\}$ is orthonormal if its vectors are mutually orthogonal and normalized:

Orthogonality: $\left\langle u_i \middle| u_j \right\rangle = 0$ for $i \neq j$.
Normalization: $\left\langle u_i \middle| u_i \right\rangle = 1$ for all $i$.

These conditions are compactly expressed using the Kronecker delta $\delta_{ij}$: \[\left\langle u_i \middle| u_j \right\rangle = \delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}\] Orthonormal bases are particularly useful in quantum mechanics for simplifying calculations and providing convenient representations of quantum states.

Hilbert Space and Linear Operators

Introduction to Hilbert Space

Hilbert space is the fundamental mathematical structure in quantum mechanics, providing the framework to describe quantum states and operators. It is a specialized vector space with additional properties that are crucial for quantum theory.

Definition of Hilbert Space

Hilbert Space: A Hilbert space $\mathcal{H}$ is a vector space over the complex numbers $\mathbb{C}$ equipped with an inner product $\left\langle \cdot \middle| \cdot \right\rangle$ that satisfies the following properties:

Linear Space: $\mathcal{H}$ is a linear space over the scalar field $\mathbb{C}$. For any vectors $\left| \psi_1 \right\rangle, \left| \psi_2 \right\rangle \in \mathcal{H}$ and scalars $c_1, c_2 \in \mathbb{C}$, the linear combination $c_1\left| \psi_1 \right\rangle + c_2\left| \psi_2 \right\rangle$ is also in $\mathcal{H}$.
Inner Product: $\mathcal{H}$ is equipped with an inner product $\left\langle \cdot \middle| \cdot \right\rangle$ (as defined in [sec:inner_product_orthogonality]) that for vectors $\left| \phi \right\rangle, \left| \psi \right\rangle, \left| \chi \right\rangle \in \mathcal{H}$ and scalar $c \in \mathbb{C}$ satisfies:
1. $\left\langle \psi \middle| \psi \right\rangle \geq 0$, and $\left\langle \psi \middle| \psi \right\rangle = 0$ if and only if $\left| \psi \right\rangle = \left| 0 \right\rangle$. (Positive-definiteness)
2. $\left\langle \phi \middle| \psi \right\rangle = \left\langle \psi \middle| \phi \right\rangle^*$. (Conjugate Symmetry)
3. $\left\langle \phi \middle| c\left| \psi \right\rangle + \left| \chi \right\rangle \right\rangle = c\left\langle \phi \middle| \psi \right\rangle + \left\langle \phi \middle| \chi \right\rangle$. (Linearity in the second argument)
Completeness: $\mathcal{H}$ is complete with respect to the norm induced by the inner product. This means every Cauchy sequence of vectors in $\mathcal{H}$ converges to a vector in $\mathcal{H}$. A sequence $\{\left| \psi_n \right\rangle\}_{n=1}^\infty$ is Cauchy if for every $\epsilon > 0$, there exists an integer $N$ such that for all $m, n > N$, $\left\| \left| \psi_m \right\rangle - \left| \psi_n \right\rangle \right\| < \epsilon$, where $\left\| \left| \psi \right\rangle \right\| = \sqrt{\left\langle \psi \middle| \psi \right\rangle}$. Completeness ensures that the space is in a sense "continuous".
Separability: $\mathcal{H}$ is separable, meaning it contains a countable dense subset. There exists a countable set of vectors $\{\left| \phi_n \right\rangle\}_{n \in \mathbb{N}} \subset \mathcal{H}$ such that any vector in $\mathcal{H}$ can be arbitrarily closely approximated by a vector from this countable set. This condition is particularly relevant for infinite-dimensional Hilbert spaces but is naturally satisfied for finite-dimensional spaces, such as those used to describe qubits.

Norm and Cauchy-Schwarz Inequality

The norm of a vector $\left| \psi \right\rangle$ in a Hilbert space $\mathcal{H}$ is defined using the inner product: \[\left\| \left| \psi \right\rangle \right\| = \sqrt{\left\langle \psi \middle| \psi \right\rangle}\] This norm is always a non-negative real number, representing the length or magnitude of the vector. It satisfies properties such as $\left\| c\left| \psi \right\rangle \right\| = |c|\left\| \left| \psi \right\rangle \right\|$ for any scalar $c \in \mathbb{C}$, and the triangle inequality $\left\| \left| \psi \right\rangle + \left| \phi \right\rangle \right\| \leq \left\| \left| \psi \right\rangle \right\| + \left\| \left| \phi \right\rangle \right\|$.

Theorem 1 (Cauchy-Schwarz Inequality). Statement of the Cauchy-Schwarz Inequality. For any vectors $\left| \psi \right\rangle, \left| \phi \right\rangle$ in a Hilbert space $\mathcal{H}$, the absolute value of their inner product is less than or equal to the product of their norms: \[|\left\langle \psi \middle| \phi \right\rangle| \leq \left\| \left| \psi \right\rangle \right\| \left\| \left| \phi \right\rangle \right\|\]

The Cauchy-Schwarz inequality is a cornerstone in the theory of Hilbert spaces and has broad applications in mathematics and physics, providing a bound on the correlation between two vectors.

Completeness and Separability in Quantum Mechanics

Completeness is a crucial property for the mathematical consistency of Hilbert spaces, ensuring that convergent sequences have limits within the space. Separability, while technically necessary for Hilbert space definition, is less emphasized in introductory quantum mechanics, especially when dealing with finite-dimensional systems relevant to qubits and quantum computing. In finite-dimensional Hilbert spaces, both completeness and separability are inherently satisfied, simplifying the mathematical treatment for many quantum systems.

Linear Operators

Linear operators act on vectors within a Hilbert space, transforming one vector into another while preserving the linear structure of the space. They are essential for describing physical transformations and measurements in quantum mechanics.

Definition of Linear Operators

Linear Operator: A linear operator $A$ on a Hilbert space $\mathcal{H}$ is a mapping $A: \mathcal{H} \to \mathcal{H}$ such that for any vectors $\left| \psi_1 \right\rangle, \left| \psi_2 \right\rangle \in \mathcal{H}$ and scalars $c_1, c_2 \in \mathbb{C}$, it satisfies the linearity condition: \[A(c_1\left| \psi_1 \right\rangle + c_2\left| \psi_2 \right\rangle) = c_1A\left| \psi_1 \right\rangle + c_2A\left| \psi_2 \right\rangle\]

Linearity is a fundamental requirement in quantum mechanics, ensuring that the evolution and manipulation of quantum states, including superpositions, are well-defined and predictable.

Examples of Linear Operators

Several key linear operators are frequently used in quantum mechanics:

Identity Operator

The identity operator $I$ (or $\mathbb{1}$) leaves any vector unchanged when applied: \[I\left| \psi \right\rangle = \left| \psi \right\rangle \quad \text{for all } \left| \psi \right\rangle \in \mathcal{H}\] In matrix form, the identity operator is represented by the identity matrix.

Zero Operator

The zero operator $0$ maps every vector in the Hilbert space to the zero vector $\left| 0 \right\rangle$: \[0\left| \psi \right\rangle = \left| 0 \right\rangle \quad \text{for all } \left| \psi \right\rangle \in \mathcal{H}\]

Projector Operator

Projector operators are crucial for describing measurements in quantum mechanics. They project a quantum state onto a specific subspace.

Definition and Action

Projector Operator: Given a normalized vector $\left| u \right\rangle \in \mathcal{H}$, the projector operator $P_{\left| u \right\rangle}$ onto the subspace spanned by $\left| u \right\rangle$ is defined as: \[P_{\left| u \right\rangle} = \left| u \right\rangle\left\langle u \right|\]

Applying$P_{\left| u \right\rangle}$ to any vector $\left| \psi \right\rangle \in \mathcal{H}$ projects $\left| \psi \right\rangle$ onto the direction of $\left| u \right\rangle$: \[P_{\left| u \right\rangle}\left| \psi \right\rangle = (\left| u \right\rangle\left\langle u \right|)\left| \psi \right\rangle = \left| u \right\rangle(\left\langle u \middle| \psi \right\rangle) = \left\langle u \middle| \psi \right\rangle\left| u \right\rangle\] The result is a vector in the direction of $\left| u \right\rangle$, scaled by the scalar value $\left\langle u \middle| \psi \right\rangle$, which is the inner product of $\left| u \right\rangle$ and $\left| \psi \right\rangle$.

Example: Projection onto Qubit States

Example 1 (Projection onto Qubit States). Consider the computational basis states for a qubit: $\left| 0 \right\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\left| 1 \right\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$. The projector onto the state $\left| 0 \right\rangle$ is $P_{\left| 0 \right\rangle} = \left| 0 \right\rangle\left\langle 0 \right|$. In matrix representation: \[P_{\left| 0 \right\rangle} = \left| 0 \right\rangle\left\langle 0 \right| = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \begin{pmatrix} 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\] Applying $P_{\left| 0 \right\rangle}$ to the state $\left| + \right\rangle = \frac{1}{\sqrt{2}}(\left| 0 \right\rangle + \left| 1 \right\rangle) = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}$: \[\begin{aligned}P_{\left| 0 \right\rangle}\left| + \right\rangle &= \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix} \\&= \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \cdot 1 + 0 \cdot 1 \\ 0 \cdot 1 + 0 \cdot 1 \end{pmatrix} \\&= \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac{1}{\sqrt{2}}\left| 0 \right\rangle\end{aligned}\] This demonstrates that projecting $\left| + \right\rangle$ onto $\left| 0 \right\rangle$ yields a vector in the $\left| 0 \right\rangle$ direction, with a scaled amplitude.

Idempotency of Projectors

Theorem 2 (Idempotency of Projectors). Projector operators are idempotent. Applying the projector twice is equivalent to applying it once: $P_{\left| u \right\rangle}^2 = P_{\left| u \right\rangle}$. \[\begin{aligned}P_{\left| u \right\rangle}^2 &= P_{\left| u \right\rangle}P_{\left| u \right\rangle} \\&= (\left| u \right\rangle\left\langle u \right|)(\ket{u\left\langle u \right|) \\&= \left| u \right\rangle(\left\langle u \middle| u \right\rangle)\left\langle u \right|\end{aligned}\] Since $\left| u \right\rangle$ is normalized, $\left\langle u \middle| u \right\rangle = 1$, thus: \[P_{\left| u \right\rangle}^2 = \left| u \right\rangle(1)\left\langle u \right| = \left| u \right\rangle\left\langle u \right| = P_{\left| u \right\rangle}\] This idempotency reflects the property that once a vector is projected onto a subspace, further projection onto the same subspace does not alter the result.

Completeness Relation and Identity Operator

Theorem 3 (Completeness Relation). Completeness Relation: For an orthonormal basis $\{\left| u_i \right\rangle\}$ of a Hilbert space $\mathcal{H}$, the completeness relation expresses the identity operator as a sum of projectors onto the basis vectors: \[\sum_{i} P_{\left| u_i \right\rangle} = \sum_{i} \left| u_i \right\rangle\left\langle u_i \right| = I\] Proof: To verify this, consider the action of $\sum_{i} \left| u_i \right\rangle\left\langle u_i \right|$ on an arbitrary vector $\left| \psi \right\rangle$: \[\left(\sum_{i} \left| u_i \right\rangle\left\langle u_i \right|\right)\left| \psi \right\rangle = \sum_{i} \left| u_i \right\rangle\left\langle u_i \middle| \psi \right\rangle\] Since $\{\left| u_i \right\rangle\}$ forms a basis, any vector $\left| \psi \right\rangle$ can be written as $\left| \psi \right\rangle = \sum_{i} c_i\left| u_i \right\rangle$, where $c_i = \left\langle u_i \middle| \psi \right\rangle$ are the components of $\left| \psi \right\rangle$ in the basis $\{\left| u_i \right\rangle\}$. Therefore: \[\sum_{i} \left| u_i \right\rangle\left\langle u_i \middle| \psi \right\rangle = \sum_{i} \left\langle u_i \middle| \psi \right\rangle\left| u_i \right\rangle = \sum_{i} c_i\left| u_i \right\rangle = \left| \psi \right\rangle\] This confirms that $\sum_{i} P_{\left| u_i \right\rangle} = I$, demonstrating that the sum of projectors over a complete orthonormal basis indeed yields the identity operator. The completeness relation is a valuable tool for decomposing the identity operator and simplifying calculations in quantum mechanics, particularly when working with orthonormal bases.

Conclusion

This lecture provided a comprehensive introduction to key concepts in quantum mechanics. We began by examining wave reflection and transmission at interfaces, emphasizing energy conservation and phase differences. Virtual laboratory demonstrations illustrated quantum phenomena such as interference, non-destructive measurement, and the Deutsch-Jozsa algorithm. The tri-polarized lens paradox highlighted the departure of quantum measurement from classical intuition. We then transitioned to a review of linear algebra, covering vector spaces, Dirac notation, inner products, linear combinations, and bases, which are foundational for quantum mechanics. Finally, we defined Hilbert spaces and linear operators, focusing on projector operators, idempotency, and the completeness relation.

Key takeaways from this lecture include the essential role of linear algebra and Hilbert spaces in formulating quantum mechanics, the fundamentally non-classical nature of quantum measurement, and the manifestation of wave-particle duality and superposition in quantum phenomena. These concepts provide the necessary mathematical and conceptual tools for describing and analyzing quantum systems.

Future lectures will delve deeper into specific quantum operators, explore time evolution according to quantum mechanics, and introduce more advanced quantum algorithms, building upon the linear algebraic and Hilbert space framework established here.

--- title: "Lecture Notes on Quantum Mechanics: Wave Reflection, Virtual Lab, Linear Algebra and Hilbert Spaces" author: "Your Name" date: "2025-02-05" format: html: toc: true # Table of Contents toc-depth: 2 code-tools: true theme: cosmo # Or "journal" for Distill-like minimalism --- # Introduction This lecture covers fundamental concepts essential for understanding quantum mechanics. We begin by examining wave reflection and transmission at an interface, focusing on energy conservation and the phase difference between transmitted and reflected waves. We then explore a virtual quantum laboratory demonstration to visualize quantum phenomena such as interference, non-destructive measurement, and the Deutsch-Jozsa algorithm. The tri-polarized lens paradox will be discussed to highlight the distinctions between classical and quantum logic. Following this, we will recap linear algebra, introducing vector spaces, Dirac notation, inner products, linear combinations, and bases. Finally, we will define Hilbert spaces and linear operators, including projector operators and the completeness relation. This lecture aims to provide a comprehensive overview, bridging wave phenomena with the abstract mathematical framework of quantum mechanics. # Wave Reflection and Transmission ## Reflection and Transmission at an Interface When a ray of light encounters an interface between two media, it is partially reflected and partially transmitted. Consider an incident wave described by the wave function $\Psi_0(x,t) = \cos(\omega t - kx)$. Upon interaction with the interface, this wave gives rise to a reflected wave $\Psi_R(x,t)$ and a transmitted wave $\Psi_T(x,t)$, both oscillating at the same frequency $\omega$. Changes in intensity and phase may occur for both the reflected and transmitted waves. <figure id="fig:reflection_transmission"> <figcaption>Reflection and Transmission at an Interface</figcaption> </figure> ## Energy Conservation and Intensity Wave intensity is proportional to the square of the wave function amplitude and represents energy flux. Assuming energy conservation at the interface, the sum of the reflected and transmitted intensities must equal the incident intensity. Let $I_0$, $I_R$, and $I_T$ denote the intensities of the incident, reflected, and transmitted waves, respectively. Energy conservation dictates: $$I_R + I_T = I_0$$ Since intensity is proportional to $|\Psi|^2$, we can express this in terms of wave function amplitudes $\Psi_0$, $\Psi_R$, and $\Psi_T$: $$|\Psi_R|^2 + |\Psi_T|^2 = |\Psi_0|^2$$ Normalizing by $|\Psi_0|^2$, and defining amplitude reflection coefficient $r = \frac{|\Psi_R|}{|\Psi_0|}$ and amplitude transmission coefficient $t = \frac{|\Psi_T|}{|\Psi_0|}$, we obtain: $$r^2 + t^2 = 1$$ This relationship expresses energy conservation in terms of amplitude coefficients. ## Phase Difference from Symmetric Two-Ray System To illustrate the phase difference between transmitted and reflected waves, consider a symmetric system with two identical incoming rays, as proposed by Didin Chain or De George. ### Symmetric Setup and Superposition Imagine two identical incoming rays with the same intensity and phase incident on a symmetric beam splitter. Each ray is partially reflected and transmitted. Due to the symmetry and superposition principle, the output wave function amplitude $\Psi_{out}$ is proportional to $(T+R)\Psi_0$, where $\Psi_0$ is the incoming wave amplitude, and $T$ and $R$ are the transmission and reflection coefficients respectively. The output intensity $I_{out} \propto |\Psi_{out}|^2 = |\Psi_0|^2 |T+R|^2$. <figure id="fig:symmetric_two_ray"> <figcaption>Symmetric Two-Ray System</figcaption> </figure> ### Phase Difference Derivation Assuming a 50/50 beam splitter and equating output intensity considerations with energy conservation, we can impose the condition $|T+R|^2 = |T|^2 + |R|^2$. Expanding the left side: $$\begin{aligned}|T+R|^2 &= (T+R)(T^*+R^*) \\&= TT^* + TR^* + RT^* + RR^* \\&= |T|^2 + |R|^2 + TR^* + RT^*\end{aligned}$$ For this to equal $|T|^2 + |R|^2$, we require $TR^* + RT^* = 0$, which simplifies to $2 \Re(TR^*) = 0$. Let $T = |T|e^{i\phi_T}$ and $R = |R|e^{i\phi_R}$. Then $TR^* = |T||R|e^{i(\phi_T - \phi_R)}$, and: $$2 \Re(|T||R|e^{i(\phi_T - \phi_R)}) = 2|T||R|\cos(\phi_T - \phi_R) = 0$$ This implies $\cos(\phi_T - \phi_R) = 0$, meaning the phase difference $\phi_T - \phi_R$ is an odd multiple of $\pi/2$. The simplest non-trivial case is $\phi_T - \phi_R = \pm \frac{\pi}{2}$. ### Phase Shift and Imaginary Unit Choosing $\phi_T - \phi_R = \frac{\pi}{2}$, we have $e^{i(\phi_T - \phi_R)} = e^{i\pi/2} = i$. This indicates a $\pi/2$ phase difference between transmitted and reflected waves. We can represent this by setting $R = i|R|$ and $T = |T|$ (or vice versa). This phase shift is fundamentally linked to the imaginary unit $i$, which is intrinsic to the time-dependent Schrödinger equation and wave phenomena in quantum mechanics. Even in simplified models, neglecting complex boundary conditions and Maxwell's equations, energy conservation arguments lead to this phase difference, highlighting the fundamental role of the imaginary unit in describing wave behavior. # Virtual Quantum Laboratory Demonstration ## Introduction to the Virtual Lab Environment A virtual quantum laboratory, accessible online, offers an interactive platform for simulating and visualizing quantum experiments. This environment allows users to manipulate quantum optical elements such as mirrors, beam splitters, and polarizers to explore phenomena like interference, non-destructive measurement, and quantum algorithms. The user interface features a selection of quantum components, sources, and detectors, enabling the setup and execution of experiments with real-time probability and outcome displays after repeated runs. Users can typically search for this \"virtual laboratory\" online to access the platform and begin experimenting. ## Interference Demonstration One of the primary demonstrations within the virtual lab is interference. By configuring an interferometer, such as a Mach-Zehnder interferometer, the wave nature of photons becomes directly observable. A typical setup involves a photon source directed into a beam splitter, which separates the photon's path into two arms. These paths are subsequently recombined at another beam splitter, with detectors positioned at the output ports to measure photon detection probabilities. By precisely adjusting the path lengths, constructive interference can be achieved at one detector, resulting in a 100% photon detection rate at that port, while the other detector registers no detections due to destructive interference. This vividly demonstrates the principles of superposition and the wave-like behavior of quantum particles. ## Non-Destructive Measurement The virtual lab also facilitates demonstrations of non-destructive measurement. By employing a specialized detector that interacts weakly with a photon, it is possible to gain path information without absorbing or destroying the photon. Observing the photon in one path in such a setup can demonstrably alter the interference pattern at the output detectors. This highlights a key distinction from classical physics: quantum measurement is not a passive observation but an interaction that can fundamentally change the state and behavior of the quantum system. ## Deutsch-Jozsa Algorithm Simulation The virtual lab provides a simulation of the Deutsch-Jozsa algorithm, a foundational quantum algorithm. In this virtual implementation, quantum gates are represented by optical elements. Specifically, the oracle in the Deutsch-Jozsa algorithm is represented by two distinct optical objects within the lab. While not implemented with standard Hadamard gates directly, components resembling square root of NOT gates are used in conjunction with these oracle objects. The Deutsch-Jozsa algorithm in this setting aims to determine whether a function is constant or balanced. The setup visualizes the principle of querying the oracle in superposition, effectively evaluating the function for both inputs (0 and 1) simultaneously within a single quantum query. The experimental configuration typically includes elements analogous to Hadamard gates before and after the oracle components, culminating in a measurement to determine the algorithm's output. This simulation allows users to observe how superposition and interference are harnessed to achieve a quantum speedup compared to classical approaches for this problem. ## Tri-polarized Lens Paradox The tri-polarized lens paradox illustrates a non-classical outcome of quantum measurement that contrasts sharply with classical intuition. ### Classical vs. Quantum Predictions Consider three polarizing lenses: A (vertically polarized), B (horizontally polarized), and C (polarized at $45^\circ$ to the vertical). Classically, if light passes through lens A and then lens B, no light is transmitted because a horizontal polarizer blocks vertically polarized light. If we represent vertically polarized photons as set A and horizontally polarized photons as set B, their intersection is empty ($A \cap B = \emptyset$). Classical logic dictates that inserting an intermediate filter C should not allow transmission when $A \cap B$ is already empty; thus, $A \cap C \cap B$ should also be empty. ### Quantum Measurement and State Modification Quantum mechanically, however, placing lens C between lenses A and B allows some light to pass through all three. After passing lens A, photons are vertically polarized. Upon encountering lens C ($45^\circ$), the vertical polarization state is projected onto the $45^\circ$ polarization direction. There is a non-zero probability for photons to pass lens C and become $45^\circ$ polarized. Subsequently, when these $45^\circ$ polarized photons reach lens B (horizontal), there is again a non-zero probability of transmission, resulting in horizontally polarized photons. Therefore, inserting lens C creates a transmission path through all three lenses, even though lenses A and B alone would block all light. This paradox arises because quantum measurement is not a passive selection but an active state modification. Measurement by lens C projects the photon into a $45^\circ$ polarized state, which has a non-zero component in the horizontal polarization direction, enabling transmission through lens B. This demonstrates that quantum measurements fundamentally alter quantum states, and classical logic does not universally apply to quantum phenomena. # Linear Algebra Recap ## Vector Spaces and Dirac Notation In quantum mechanics, physical states are represented as vectors within a vector space. Dirac notation, also known as bra-ket notation, is the standard convention for representing these vectors. ### Ket and Bra Vectors A vector in a complex vector space is denoted by a ket, symbolized as $\left| \psi \right\rangle$ (e.g., $\left| v \right\rangle$). In component form, a ket is represented as a column vector. The dual vector, known as a bra, is denoted as $\left\langle \psi \right|$ and is the conjugate transpose (Hermitian conjugate) of the ket vector. In component form, a bra is represented as a row vector with complex conjugated entries. For a ket vector $\left| v \right\rangle = \begin{pmatrix} c_1 \\ c_2 \\ \vdots \\ c_n \end{pmatrix}$, its corresponding bra vector is $\left\langle v \right| = \begin{pmatrix} c_1^* & c_2^* & \cdots & c_n^* \end{pmatrix}$, where $c_i \in \mathbb{C}$ and $c_i^*$ is the complex conjugate of $c_i$. ### Vector Operations Vector spaces are closed under vector addition and scalar multiplication, fundamental operations in linear algebra. - **Vector Addition:** For any two vectors $\left| u \right\rangle, \left| v \right\rangle$ in a vector space $V$, their sum $\left| u \right\rangle + \left| v \right\rangle$ is also in $V$. Vector addition is performed component-wise. - **Scalar Multiplication:** For a vector $\left| v \right\rangle \in V$ and a scalar $c \in \mathbb{C}$, the scalar product $c\left| v \right\rangle$ is also in $V$. Scalar multiplication involves multiplying each component of $\left| v \right\rangle$ by $c$. The zero vector $\left| 0 \right\rangle$ is the additive identity, such that for any vector $\left| v \right\rangle$, $\left| v \right\rangle + \left| 0 \right\rangle = \left| v \right\rangle$. Every vector $\left| v \right\rangle$ has an additive inverse $-\left| v \right\rangle$ satisfying $\left| v \right\rangle + (-\left| v \right\rangle) = \left| 0 \right\rangle$. ## Inner Product and Orthogonality The inner product, or scalar product, generalizes the dot product to complex vector spaces. It maps two vectors to a scalar and defines orthogonality. ### Definition of Inner Product ::: tcolorbox **Inner Product:** The inner product of two vectors $\left| x \right\rangle$ and $\left| y \right\rangle$ in a complex vector space, denoted as $\left\langle x \middle| y \right\rangle$, is defined as: $$\left\langle x \middle| y \right\rangle = \left\langle x \right| \left| y \right\rangle = \sum_{i} x_i^* y_i$$ where $\left| x \right\rangle = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}$ and $\left| y \right\rangle = \begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix}$. ::: ### Properties of the Inner Product The inner product satisfies the following properties for vectors $\left| x \right\rangle, \left| y \right\rangle, \left| z \right\rangle$ and scalars $c_1, c_2 \in \mathbb{C}$: 1. **Conjugate Symmetry:** $\left\langle x \middle| y \right\rangle = \left\langle y \middle| x \right\rangle^*$ 2. **Linearity in the second argument:** $\left\langle x \middle| c_1\left| y_1 \right\rangle + c_2\left| y_2 \right\rangle \right\rangle = c_1\left\langle x \middle| y_1 \right\rangle + c_2\left\langle x \middle| y_2 \right\rangle$ 3. **Conjugate Linearity in the first argument:** $\left\langle c_1\left| x_1 \right\rangle + c_2\left| x_2 \right\rangle \middle| y \right\rangle = c_1^*\left\langle x_1 \middle| y \right\rangle + c_2^*\left\langle x_2 \middle| y \right\rangle$ 4. **Positive-Definiteness:** $\left\langle x \middle| x \right\rangle \geq 0$, with $\left\langle x \middle| x \right\rangle = 0$ if and only if $\left| x \right\rangle = \left| 0 \right\rangle$. Note the non-commutativity: $\left\langle x \middle| y \right\rangle \neq \left\langle y \middle| x \right\rangle$ in general, but rather $\left\langle x \middle| y \right\rangle = \left\langle y \middle| x \right\rangle^*$. ### Orthogonality Two vectors $\left| x \right\rangle$ and $\left| y \right\rangle$ are orthogonal if their inner product is zero: $$\left\langle x \middle| y \right\rangle = 0$$ ## Linear Combinations and Linear Independence Linear combinations and linear independence are essential for characterizing vector space structure. ### Linear Combinations and Linear Manifolds (Subspaces) A linear combination of vectors $\{\left| v_1 \right\rangle, \left| v_2 \right\rangle, \ldots, \left| v_n \right\rangle\}$ is a vector $\left| v \right\rangle$ expressed as: $$\left| v \right\rangle = c_1\left| v_1 \right\rangle + c_2\left| v_2 \right\rangle + \cdots + c_n\left| v_n \right\rangle = \sum_{i=1}^{n} c_i\left| v_i \right\rangle$$ where $c_i \in \mathbb{C}$ are scalar coefficients. The set of all possible linear combinations of $\{\left| v_1 \right\rangle, \left| v_2 \right\rangle, \ldots, \left| v_n \right\rangle\}$ forms a linear manifold, or subspace, spanned by these vectors. ### Linear Dependence and Independence A set of vectors $\{\left| v_1 \right\rangle, \left| v_2 \right\rangle, \ldots, \left| v_n \right\rangle\}$ is linearly dependent if there exist coefficients $\{c_1, c_2, \ldots, c_n\}$, not all zero, such that: $$\sum_{i=1}^{n} c_i\left| v_i \right\rangle = \left| 0 \right\rangle$$ If the only solution is $c_1 = c_2 = \cdots = c_n = 0$, the set is linearly independent. Equivalently, linear dependence means at least one vector can be expressed as a linear combination of the others; otherwise, they are linearly independent. ### Checking Linear Independence Linear independence can be checked using methods like Gaussian elimination. Consider vectors $\left| v_1 \right\rangle = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix}$, $\left| v_2 \right\rangle = \begin{pmatrix} 4 \\ 7 \\ 4 \end{pmatrix}$, $\left| v_3 \right\rangle = \begin{pmatrix} 2 \\ 5 \\ 3 \end{pmatrix}$. We transform them using linear combinations: 1. Keep $\left| v_1 \right\rangle$ as is: $\left| v_1 \right\rangle = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix}$. 2. Replace $\left| v_2 \right\rangle$ with $\left| v_2' \right\rangle = \left| v_2 \right\rangle - 4\left| v_1 \right\rangle = \begin{pmatrix} 0 \\ -5 \\ -4 \end{pmatrix}$. 3. Replace $\left| v_3 \right\rangle$ with $\left| v_3' \right\rangle = \left| v_3 \right\rangle - 2\left| v_1 \right\rangle = \begin{pmatrix} 0 \\ -1 \\ -1 \end{pmatrix}$. 4. Replace $\left| v_3' \right\rangle$ with $\left| v_3'' \right\rangle = \left| v_3' \right\rangle - \frac{1}{5}\left| v_2' \right\rangle = \begin{pmatrix} 0 \\ 0 \\ -1/5 \end{pmatrix}$. The transformed set is $\left| v_1 \right\rangle = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix}$, $\left| v_2' \right\rangle = \begin{pmatrix} 0 \\ -5 \\ -4 \end{pmatrix}$, $\left| v_3'' \right\rangle = \begin{pmatrix} 0 \\ 0 \\ -1/5 \end{pmatrix}$. Since we have non-zero components along different dimensions after transformation, the original vectors are linearly independent. If a zero vector were obtained during this process, it would indicate linear dependence. ## Basis and Orthonormal Basis A basis is a minimal set of vectors that spans the entire vector space. An orthonormal basis is a basis composed of orthogonal and normalized vectors, which simplifies many calculations. ### Definition of a Basis ::: tcolorbox **Basis:** A set of vectors $\{\left| b_1 \right\rangle, \left| b_2 \right\rangle, \ldots, \left| b_n \right\rangle\}$ is a basis for a vector space $V$ if it satisfies two conditions: 1. **Linear Independence:** $\{\left| b_1 \right\rangle, \left| b_2 \right\rangle, \ldots, \left| b_n \right\rangle\}$ is a linearly independent set. 2. **Spanning:** $\{\left| b_1 \right\rangle, \left| b_2 \right\rangle, \ldots, \left| b_n \right\rangle\}$ spans $V$, meaning any vector $\left| v \right\rangle \in V$ can be expressed as a linear combination of $\{\left| b_1 \right\rangle, \left| b_2 \right\rangle, \ldots, \left| b_n \right\rangle\}$. In an $n$-dimensional vector space, any set of $n$ linearly independent vectors constitutes a basis. ::: ### Orthonormal Basis and Kronecker Delta A basis $\{\left| u_1 \right\rangle, \left| u_2 \right\rangle, \ldots, \left| u_n \right\rangle\}$ is orthonormal if its vectors are mutually orthogonal and normalized: 1. **Orthogonality:** $\left\langle u_i \middle| u_j \right\rangle = 0$ for $i \neq j$. 2. **Normalization:** $\left\langle u_i \middle| u_i \right\rangle = 1$ for all $i$. These conditions are compactly expressed using the Kronecker delta $\delta_{ij}$: $$\left\langle u_i \middle| u_j \right\rangle = \delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}$$ Orthonormal bases are particularly useful in quantum mechanics for simplifying calculations and providing convenient representations of quantum states. # Hilbert Space and Linear Operators ## Introduction to Hilbert Space Hilbert space is the fundamental mathematical structure in quantum mechanics, providing the framework to describe quantum states and operators. It is a specialized vector space with additional properties that are crucial for quantum theory. ### Definition of Hilbert Space ::: tcolorbox **Hilbert Space:** A Hilbert space $\mathcal{H}$ is a vector space over the complex numbers $\mathbb{C}$ equipped with an inner product $\left\langle \cdot \middle| \cdot \right\rangle$ that satisfies the following properties: 1. **Linear Space:** $\mathcal{H}$ is a linear space over the scalar field $\mathbb{C}$. For any vectors $\left| \psi_1 \right\rangle, \left| \psi_2 \right\rangle \in \mathcal{H}$ and scalars $c_1, c_2 \in \mathbb{C}$, the linear combination $c_1\left| \psi_1 \right\rangle + c_2\left| \psi_2 \right\rangle$ is also in $\mathcal{H}$. 2. **Inner Product:** $\mathcal{H}$ is equipped with an inner product $\left\langle \cdot \middle| \cdot \right\rangle$ (as defined in [\[sec:inner_product_orthogonality\]](#sec:inner_product_orthogonality){reference-type="ref+label" reference="sec:inner_product_orthogonality"}) that for vectors $\left| \phi \right\rangle, \left| \psi \right\rangle, \left| \chi \right\rangle \in \mathcal{H}$ and scalar $c \in \mathbb{C}$ satisfies: 1. $\left\langle \psi \middle| \psi \right\rangle \geq 0$, and $\left\langle \psi \middle| \psi \right\rangle = 0$ if and only if $\left| \psi \right\rangle = \left| 0 \right\rangle$. (Positive-definiteness) 2. $\left\langle \phi \middle| \psi \right\rangle = \left\langle \psi \middle| \phi \right\rangle^*$. (Conjugate Symmetry) 3. $\left\langle \phi \middle| c\left| \psi \right\rangle + \left| \chi \right\rangle \right\rangle = c\left\langle \phi \middle| \psi \right\rangle + \left\langle \phi \middle| \chi \right\rangle$. (Linearity in the second argument) 3. **Completeness:** $\mathcal{H}$ is complete with respect to the norm induced by the inner product. This means every Cauchy sequence of vectors in $\mathcal{H}$ converges to a vector in $\mathcal{H}$. A sequence $\{\left| \psi_n \right\rangle\}_{n=1}^\infty$ is Cauchy if for every $\epsilon > 0$, there exists an integer $N$ such that for all $m, n > N$, $\left\| \left| \psi_m \right\rangle - \left| \psi_n \right\rangle \right\| < \epsilon$, where $\left\| \left| \psi \right\rangle \right\| = \sqrt{\left\langle \psi \middle| \psi \right\rangle}$. Completeness ensures that the space is in a sense \"continuous\". 4. **Separability:** $\mathcal{H}$ is separable, meaning it contains a countable dense subset. There exists a countable set of vectors $\{\left| \phi_n \right\rangle\}_{n \in \mathbb{N}} \subset \mathcal{H}$ such that any vector in $\mathcal{H}$ can be arbitrarily closely approximated by a vector from this countable set. This condition is particularly relevant for infinite-dimensional Hilbert spaces but is naturally satisfied for finite-dimensional spaces, such as those used to describe qubits. ::: ### Norm and Cauchy-Schwarz Inequality The norm of a vector $\left| \psi \right\rangle$ in a Hilbert space $\mathcal{H}$ is defined using the inner product: $$\left\| \left| \psi \right\rangle \right\| = \sqrt{\left\langle \psi \middle| \psi \right\rangle}$$ This norm is always a non-negative real number, representing the length or magnitude of the vector. It satisfies properties such as $\left\| c\left| \psi \right\rangle \right\| = |c|\left\| \left| \psi \right\rangle \right\|$ for any scalar $c \in \mathbb{C}$, and the triangle inequality $\left\| \left| \psi \right\rangle + \left| \phi \right\rangle \right\| \leq \left\| \left| \psi \right\rangle \right\| + \left\| \left| \phi \right\rangle \right\|$. ::: theorem **Theorem 1** (Cauchy-Schwarz Inequality). ***Statement of the Cauchy-Schwarz Inequality.** For any vectors $\left| \psi \right\rangle, \left| \phi \right\rangle$ in a Hilbert space $\mathcal{H}$, the absolute value of their inner product is less than or equal to the product of their norms: $$|\left\langle \psi \middle| \phi \right\rangle| \leq \left\| \left| \psi \right\rangle \right\| \left\| \left| \phi \right\rangle \right\|$$* ::: The Cauchy-Schwarz inequality is a cornerstone in the theory of Hilbert spaces and has broad applications in mathematics and physics, providing a bound on the correlation between two vectors. ### Completeness and Separability in Quantum Mechanics Completeness is a crucial property for the mathematical consistency of Hilbert spaces, ensuring that convergent sequences have limits within the space. Separability, while technically necessary for Hilbert space definition, is less emphasized in introductory quantum mechanics, especially when dealing with finite-dimensional systems relevant to qubits and quantum computing. In finite-dimensional Hilbert spaces, both completeness and separability are inherently satisfied, simplifying the mathematical treatment for many quantum systems. ## Linear Operators Linear operators act on vectors within a Hilbert space, transforming one vector into another while preserving the linear structure of the space. They are essential for describing physical transformations and measurements in quantum mechanics. ### Definition of Linear Operators ::: tcolorbox **Linear Operator:** A linear operator $A$ on a Hilbert space $\mathcal{H}$ is a mapping $A: \mathcal{H} \to \mathcal{H}$ such that for any vectors $\left| \psi_1 \right\rangle, \left| \psi_2 \right\rangle \in \mathcal{H}$ and scalars $c_1, c_2 \in \mathbb{C}$, it satisfies the linearity condition: $$A(c_1\left| \psi_1 \right\rangle + c_2\left| \psi_2 \right\rangle) = c_1A\left| \psi_1 \right\rangle + c_2A\left| \psi_2 \right\rangle$$ ::: Linearity is a fundamental requirement in quantum mechanics, ensuring that the evolution and manipulation of quantum states, including superpositions, are well-defined and predictable. ### Examples of Linear Operators Several key linear operators are frequently used in quantum mechanics: #### Identity Operator The identity operator $I$ (or $\mathbb{1}$) leaves any vector unchanged when applied: $$I\left| \psi \right\rangle = \left| \psi \right\rangle \quad \text{for all } \left| \psi \right\rangle \in \mathcal{H}$$ In matrix form, the identity operator is represented by the identity matrix. #### Zero Operator The zero operator $0$ maps every vector in the Hilbert space to the zero vector $\left| 0 \right\rangle$: $$0\left| \psi \right\rangle = \left| 0 \right\rangle \quad \text{for all } \left| \psi \right\rangle \in \mathcal{H}$$ #### Projector Operator Projector operators are crucial for describing measurements in quantum mechanics. They project a quantum state onto a specific subspace. ##### Definition and Action ::: tcolorbox **Projector Operator:** Given a normalized vector $\left| u \right\rangle \in \mathcal{H}$, the projector operator $P_{\left| u \right\rangle}$ onto the subspace spanned by $\left| u \right\rangle$ is defined as: $$P_{\left| u \right\rangle} = \left| u \right\rangle\left\langle u \right|$$ ::: Applying$P_{\left| u \right\rangle}$ to any vector $\left| \psi \right\rangle \in \mathcal{H}$ projects $\left| \psi \right\rangle$ onto the direction of $\left| u \right\rangle$: $$P_{\left| u \right\rangle}\left| \psi \right\rangle = (\left| u \right\rangle\left\langle u \right|)\left| \psi \right\rangle = \left| u \right\rangle(\left\langle u \middle| \psi \right\rangle) = \left\langle u \middle| \psi \right\rangle\left| u \right\rangle$$ The result is a vector in the direction of $\left| u \right\rangle$, scaled by the scalar value $\left\langle u \middle| \psi \right\rangle$, which is the inner product of $\left| u \right\rangle$ and $\left| \psi \right\rangle$. ##### Example: Projection onto Qubit States ::: example **Example 1** (Projection onto Qubit States). Consider the computational basis states for a qubit: $\left| 0 \right\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\left| 1 \right\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$. The projector onto the state $\left| 0 \right\rangle$ is $P_{\left| 0 \right\rangle} = \left| 0 \right\rangle\left\langle 0 \right|$. In matrix representation: $$P_{\left| 0 \right\rangle} = \left| 0 \right\rangle\left\langle 0 \right| = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \begin{pmatrix} 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$$ Applying $P_{\left| 0 \right\rangle}$ to the state $\left| + \right\rangle = \frac{1}{\sqrt{2}}(\left| 0 \right\rangle + \left| 1 \right\rangle) = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}$: $$\begin{aligned}P_{\left| 0 \right\rangle}\left| + \right\rangle &= \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix} \\&= \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \cdot 1 + 0 \cdot 1 \\ 0 \cdot 1 + 0 \cdot 1 \end{pmatrix} \\&= \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac{1}{\sqrt{2}}\left| 0 \right\rangle\end{aligned}$$ This demonstrates that projecting $\left| + \right\rangle$ onto $\left| 0 \right\rangle$ yields a vector in the $\left| 0 \right\rangle$ direction, with a scaled amplitude. ::: ##### Idempotency of Projectors ::: theorem **Theorem 2** (Idempotency of Projectors). ***Projector operators are idempotent.** Applying the projector twice is equivalent to applying it once: $P_{\left| u \right\rangle}^2 = P_{\left| u \right\rangle}$. $$\begin{aligned}P_{\left| u \right\rangle}^2 &= P_{\left| u \right\rangle}P_{\left| u \right\rangle} \\&= (\left| u \right\rangle\left\langle u \right|)(\ket{u\left\langle u \right|) \\&= \left| u \right\rangle(\left\langle u \middle| u \right\rangle)\left\langle u \right|\end{aligned}$$ Since $\left| u \right\rangle$ is normalized, $\left\langle u \middle| u \right\rangle = 1$, thus: $$P_{\left| u \right\rangle}^2 = \left| u \right\rangle(1)\left\langle u \right| = \left| u \right\rangle\left\langle u \right| = P_{\left| u \right\rangle}$$ This idempotency reflects the property that once a vector is projected onto a subspace, further projection onto the same subspace does not alter the result.* ::: ### Completeness Relation and Identity Operator ::: theorem **Theorem 3** (Completeness Relation). ***Completeness Relation:** For an orthonormal basis $\{\left| u_i \right\rangle\}$ of a Hilbert space $\mathcal{H}$, the completeness relation expresses the identity operator as a sum of projectors onto the basis vectors: $$\sum_{i} P_{\left| u_i \right\rangle} = \sum_{i} \left| u_i \right\rangle\left\langle u_i \right| = I$$ **Proof:** To verify this, consider the action of $\sum_{i} \left| u_i \right\rangle\left\langle u_i \right|$ on an arbitrary vector $\left| \psi \right\rangle$: $$\left(\sum_{i} \left| u_i \right\rangle\left\langle u_i \right|\right)\left| \psi \right\rangle = \sum_{i} \left| u_i \right\rangle\left\langle u_i \middle| \psi \right\rangle$$ Since $\{\left| u_i \right\rangle\}$ forms a basis, any vector $\left| \psi \right\rangle$ can be written as $\left| \psi \right\rangle = \sum_{i} c_i\left| u_i \right\rangle$, where $c_i = \left\langle u_i \middle| \psi \right\rangle$ are the components of $\left| \psi \right\rangle$ in the basis $\{\left| u_i \right\rangle\}$. Therefore: $$\sum_{i} \left| u_i \right\rangle\left\langle u_i \middle| \psi \right\rangle = \sum_{i} \left\langle u_i \middle| \psi \right\rangle\left| u_i \right\rangle = \sum_{i} c_i\left| u_i \right\rangle = \left| \psi \right\rangle$$ This confirms that $\sum_{i} P_{\left| u_i \right\rangle} = I$, demonstrating that the sum of projectors over a complete orthonormal basis indeed yields the identity operator. The completeness relation is a valuable tool for decomposing the identity operator and simplifying calculations in quantum mechanics, particularly when working with orthonormal bases.* ::: []{#sec:inner_product_orthogonality label="sec:inner_product_orthogonality"} # Conclusion This lecture provided a comprehensive introduction to key concepts in quantum mechanics. We began by examining wave reflection and transmission at interfaces, emphasizing energy conservation and phase differences. Virtual laboratory demonstrations illustrated quantum phenomena such as interference, non-destructive measurement, and the Deutsch-Jozsa algorithm. The tri-polarized lens paradox highlighted the departure of quantum measurement from classical intuition. We then transitioned to a review of linear algebra, covering vector spaces, Dirac notation, inner products, linear combinations, and bases, which are foundational for quantum mechanics. Finally, we defined Hilbert spaces and linear operators, focusing on projector operators, idempotency, and the completeness relation. Key takeaways from this lecture include the essential role of linear algebra and Hilbert spaces in formulating quantum mechanics, the fundamentally non-classical nature of quantum measurement, and the manifestation of wave-particle duality and superposition in quantum phenomena. These concepts provide the necessary mathematical and conceptual tools for describing and analyzing quantum systems. Future lectures will delve deeper into specific quantum operators, explore time evolution according to quantum mechanics, and introduce more advanced quantum algorithms, building upon the linear algebraic and Hilbert space framework established here.