Lecture Notes on Linear Algebra and Introduction to Quantum Mechanics

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February 5, 2025

Introduction

This lecture reviews matrix diagonalization and explores linear algebra exercises relevant to quantum mechanics. It then transitions into an introduction to quantum mechanics, discussing its motivations, early developments, formulations, and fundamental postulates. The lecture concludes by connecting quantum and classical mechanics through the correspondence principle and introducing the Schrödinger equation.

Linear Algebra Exercises

Example of a Non-Diagonalizable Matrix

We aim to demonstrate that the matrix \[M = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}\] cannot be diagonalized.

To determine if $M$ is diagonalizable, we first find its eigenvalues by solving the characteristic equation $\det(M - \lambda I) = 0$, where $I$ is the $2 \times 2$ identity matrix and $\lambda$ represents the eigenvalues. \[\begin{aligned}\det(M - \lambda I) &= \det \left( \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} - \lambda \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \right) \\&= \det \begin{pmatrix} 1-\lambda & 0 \\ 1 & 1-\lambda \end{pmatrix} \\&= (1-\lambda)(1-\lambda) - (0)(1) \\&= (1-\lambda)^2\end{aligned}\] Setting the determinant to zero, $(1-\lambda)^2 = 0$, we find a repeated eigenvalue $\lambda = 1$ with algebraic multiplicity 2.

Next, we find the eigenvectors corresponding to $\lambda = 1$ by solving the homogeneous system $(M - \lambda I)v = 0$: \[\begin{pmatrix} 1-1 & 0 \\ 1 & 1-1 \end{pmatrix} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}\] \[\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}\] This matrix equation is equivalent to the system of linear equations: \[\begin{aligned}0\alpha + 0\beta &= 0 \\1\alpha + 0\beta &= 0\end{aligned}\] From the second equation, we have $\alpha = 0$. The variable $\beta$ is a free variable, meaning it can be any complex number. Choosing $\beta = 1$, we obtain an eigenvector $v = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$. Thus, all eigenvectors are of the form $c \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ for any non-zero scalar $c$.

The geometric multiplicity of the eigenvalue $\lambda = 1$ is the dimension of the eigenspace, which is 1 (since all eigenvectors are scalar multiples of each other). Since the geometric multiplicity (1) is less than the algebraic multiplicity (2), the matrix $M$ is not diagonalizable.

Standard Procedure for Eigenvalues and Eigenvectors

Algorithm 1 (H).

Form the Characteristic Equation: Construct the characteristic equation $\det(A - \lambda I) = 0$, where $I$ is the identity matrix of the same dimension as $A$, and $\lambda$ represents the eigenvalues. Solve for Eigenvalues: Solve the characteristic equation for $\lambda$. The roots $\lambda_1, \lambda_2, \dots, \lambda_n$ of this polynomial are the eigenvalues of $A$. Determine Eigenvectors: For each eigenvalue $\lambda_i$, solve the homogeneous system of linear equations $(A - \lambda_i I)v_i = 0$ to find the corresponding eigenvector(s) $v_i$. The set of all non-zero solutions $v_i$ forms the eigenspace associated with $\lambda_i$.

Adjoint (Dagger) of an Operator

This theorem derives the adjoint of an operator defined as the outer product of two vectors.

Proposition 1. For an operator $O = \left| U \right\rangle\left\langle W \right|$, its adjoint is $O^\dagger = \left| W \right\rangle\left\langle U \right|$.

Proof. Proof. To prove this, we use the definition of the adjoint operator in terms of its matrix elements. For any operators $A$ and $B$ and vectors $\left| \phi \right\rangle$ and $\left| \psi \right\rangle$, the adjoint is defined such that $\left\langle \phi \right| A^\dagger \left| \psi \right\rangle = (\left\langle \psi \right| A \left| \phi \right\rangle)^*$. Let $O = \left| U \right\rangle\left\langle W \right|$. Then for arbitrary vectors $\left| I \right\rangle$ and $\left| J \right\rangle$: \[\begin{aligned}\left\langle J \right| O^\dagger \left| I \right\rangle &= (\left\langle I \right| O \left| J \right\rangle)^* \\&= (\left\langle I \right| (\left| U \right\rangle\left\langle W \right|) \left| J \right\rangle)^* \\&= (\left\langle I \middle| U \right\rangle \left\langle W \middle| J \right\rangle)^* \\&= \left\langle I \middle| U \right\rangle^* \left\langle W \middle| J \right\rangle^* \\&= \left\langle U \middle| I \right\rangle \left\langle J \middle| W \right\rangle \\&= \left\langle J \middle| W \right\rangle \left\langle U \middle| I \right\rangle \\&= \left\langle J \right| (\left| W \right\rangle\left\langle U \right|) \left| I \right\rangle\end{aligned}\] Since this equality holds for all vectors $\left| I \right\rangle$ and $\left| J \right\rangle$, we conclude that the operator $O^\dagger$ is indeed $\left| W \right\rangle\left\langle U \right|$. ◻

Normal and Hermitian Operators

Definitions

An operator $A$ is Hermitian if it is equal to its adjoint, i.e., $A = A^\dagger$. Hermitian operators represent physical observables in quantum mechanics.

An operator $A$ is normal if it commutes with its adjoint, i.e., $AA^\dagger = A^\dagger A$. Normal operators are important because they are diagonalizable by a unitary transformation.

Hermitian Condition for Normal Operators

This theorem states the necessary and sufficient condition for a normal matrix to be Hermitian based on its eigenvalues.

Theorem 2. A normal matrix $A$ is Hermitian if and only if all its eigenvalues are real.

Proof. Proof. Let $A$ be a normal matrix. By the spectral theorem for normal operators, there exists a unitary matrix $U$ such that $A = UDU^\dagger$, where $D$ is a diagonal matrix whose diagonal entries are the eigenvalues of $A$. The adjoint of $A$ is $A^\dagger = (UDU^\dagger)^\dagger = (U^\dagger)^\dagger D^\dagger U^\dagger = UD^\dagger U^\dagger$.

($\Rightarrow$) Assume $A$ is Hermitian, so $A = A^\dagger$. Then $UDU^\dagger = UD^\dagger U^\dagger$. Multiplying by $U^\dagger$ on the left and $U$ on the right, we obtain $D = D^\dagger$. For a diagonal matrix, the adjoint is obtained by taking the complex conjugate of each diagonal entry. Thus, $D = D^\dagger$ implies that all diagonal entries of $D$, which are the eigenvalues of $A$, must be real.

($\Leftarrow$) Assume all eigenvalues of $A$ are real. Then the diagonal matrix $D$ has real entries, so $D = D^\dagger$. Therefore, $A^\dagger = UD^\dagger U^\dagger = UDU^\dagger = A$. Hence, $A$ is Hermitian. ◻

Spectral Resolution

A normal operator $A$ can be decomposed using its spectral resolution as: \[A = \sum_{k} \lambda_k P_k\] where $\lambda_k$ are the eigenvalues of $A$, and $P_k$ are the orthogonal projectors onto the eigenspaces corresponding to $\lambda_k$. For a Hermitian operator, the eigenvalues $\lambda_k$ are guaranteed to be real, and the eigenvectors corresponding to distinct eigenvalues are orthogonal.

Tensor Product Operations

Tensor Product of State Vectors

Given a single-qubit state $\left| \psi \right\rangle = \frac{1}{\sqrt{2}} (\left| 0 \right\rangle + \left| 1 \right\rangle)$, the tensor product of two such states is: \[\begin{aligned}\left| \psi \right\rangle \otimes \left| \psi \right\rangle &= \left( \frac{1}{\sqrt{2}} (\left| 0 \right\rangle + \left| 1 \right\rangle) \right) \otimes \left( \frac{1}{\sqrt{2}} (\left| 0 \right\rangle + \left| 1 \right\rangle) \right) \\&= \frac{1}{2} (\left| 0 \right\rangle \otimes \left| 0 \right\rangle + \left| 0 \right\rangle \otimes \left| 1 \right\rangle + \left| 1 \right\rangle \otimes \left| 0 \right\rangle + \left| 1 \right\rangle \otimes \left| 1 \right\rangle) \\&= \frac{1}{2} (\left| 00 \right\rangle + \left| 01 \right\rangle + \left| 10 \right\rangle + \left| 11 \right\rangle)\end{aligned}\] For $n$ qubits, the $n$-fold tensor product is: \[\left| \psi \right\rangle^{\otimes n} = \left( \frac{1}{\sqrt{2}} (\left| 0 \right\rangle + \left| 1 \right\rangle) \right)^{\otimes n} = \frac{1}{2^{n/2}} \sum_{x_1 \in \{0,1\}} \cdots \sum_{x_n \in \{0,1\}} \left| x_1 \cdots x_n \right\rangle = \frac{1}{2^{n/2}} \sum_{x \in \{0,1\}^n} \left| x \right\rangle\] This represents an equal superposition over all $2^n$ computational basis states, which is crucial in quantum algorithms.

Tensor Product of Operators

If an operator $A$ acts on a vector space $V_1$ and an operator $B$ acts on a vector space $V_2$, their tensor product $A \otimes B$ acts on the tensor product space $V_1 \otimes V_2$. In matrix form, if $A$ is an $m \times m$ matrix and $B$ is an $n \times n$ matrix, the tensor product $A \otimes B$ is an $mn \times mn$ matrix. If $A = (a_{ij})$ and $B = (b_{kl})$, the matrix for $A \otimes B$ can be constructed in blocks. For example, if $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$, then \[A \otimes B = \begin{pmatrix} a_{11}B & a_{12}B \\ a_{21}B & a_{22}B \end{pmatrix}\] where each block $a_{ij}B$ is a matrix obtained by multiplying each entry of $B$ by the scalar $a_{ij}$.

Transpose and Adjoint Properties of Tensor Products

This theorem demonstrates that the transpose operation distributes over the tensor product.

Proposition 3. The transpose of a tensor product is the tensor product of the transposes: $(A \otimes B)^T = A^T \otimes B^T$.

Proof. Proof. Let $A$ be an $m \times m$ matrix and $B$ be an $n \times n$ matrix. The $((i, k), (j, l))$-th element of $A \otimes B$ is given by $(A \otimes B)_{(i, k), (j, l)} = A_{ij} B_{kl}$. The transpose operation swaps rows and columns. Thus, the $((j, l), (i, k))$-th element of $(A \otimes B)^T$ is $((A \otimes B)^T)_{(j, l), (i, k)} = (A \otimes B)_{(i, k), (j, l)} = A_{ij} B_{kl}$. On the other hand, the $((j, l), (i, k))$-th element of $A^T \otimes B^T$ is $(A^T \otimes B^T)_{(j, l), (i, k)} = (A^T)_{ji} (B^T)_{lk} = A_{ij} B_{kl}$. Since the elements are equal, we conclude that $(A \otimes B)^T = A^T \otimes B^T$. ◻

This theorem demonstrates that the adjoint operation distributes over the tensor product.

Proposition 4. The adjoint of a tensor product is the tensor product of the adjoints: $(A \otimes B)^\dagger = A^\dagger \otimes B^\dagger$.

Proof. Proof. Using the property that for any matrix $M$, $(M^\dagger) = (M^T)^* = (M^*)^T$, we have: \[(A \otimes B)^\dagger = ((A \otimes B)^T)^* = (A^T \otimes B^T)^* = (A^T)^* \otimes (B^T)^* = A^\dagger \otimes B^\dagger\] ◻

This theorem shows that the transpose of a product of operators is the product of transposes in reverse order.

Proposition 5. The transpose of a product of operators is the product of transposes in reverse order: $(AB)^T = B^T A^T$.

Proof. Proof. Let $A$ and $B$ be matrices such that their product $AB$ is defined. The $(i, j)$-th element of $AB$ is $(AB)_{ij} = \sum_k A_{ik} B_{kj}$. The transpose operation swaps indices, so the $(j, i)$-th element of $(AB)^T$ is $((AB)^T)_{ji} = (AB)_{ij} = \sum_k A_{ik} B_{kj}$. Now consider the product $B^T A^T$. Its $(j, i)$-th element is $(B^T A^T)_{ji} = \sum_k (B^T)_{jk} (A^T)_{ki} = \sum_k B_{kj} A_{ik} = \sum_k A_{ik} B_{kj}$. Since $((AB)^T)_{ji} = (B^T A^T)_{ji}$ for all $i, j$, we conclude that $(AB)^T = B^T A^T$. ◻

This theorem shows that the adjoint of a product of operators is the product of adjoints in reverse order.

Proposition 6. The adjoint of a product of operators is the product of the adjoints in reverse order: $(AB)^\dagger = B^\dagger A^\dagger$.

Proof. Proof. Using the property that $(M^\dagger) = (M^T)^* = (M^*)^T$ and the transpose property of a product of operators, we have: \[(AB)^\dagger = ((AB)^T)^* = (B^T A^T)^* = (B^T)^* (A^T)^* = B^\dagger A^\dagger\] ◻

Representation of the Hadamard Operator

The Hadamard operator $H$ is a fundamental single-qubit gate in quantum computing, defined by its action on the computational basis states $\left| 0 \right\rangle$ and $\left| 1 \right\rangle$: \[\begin{aligned}H\left| 0 \right\rangle &= \frac{1}{\sqrt{2}} (\left| 0 \right\rangle + \left| 1 \right\rangle) \\H\left| 1 \right\rangle &= \frac{1}{\sqrt{2}} (\left| 0 \right\rangle - \left| 1 \right\rangle)\end{aligned}\] We can express the Hadamard operator using projectors onto the computational basis states: \[\begin{aligned}H &= H \left| 0 \right\rangle\left\langle 0 \right| + H \left| 1 \right\rangle\left\langle 1 \right| \\&= \frac{1}{\sqrt{2}} (\left| 0 \right\rangle + \left| 1 \right\rangle)\left\langle 0 \right| + \frac{1}{\sqrt{2}} (\left| 0 \right\rangle - \left| 1 \right\rangle)\left\langle 1 \right| \\&= \frac{1}{\sqrt{2}} (\left| 0 \right\rangle\left\langle 0 \right| + \left| 1 \right\rangle\left\langle 0 \right| + \left| 0 \right\rangle\left\langle 1 \right| - \left| 1 \right\rangle\left\langle 1 \right|) \\&= \frac{1}{\sqrt{2}} (\left| 0 \right\rangle\left\langle 0 \right| + \left| 0 \right\rangle\left\langle 1 \right| + \left| 1 \right\rangle\left\langle 0 \right| - \left| 1 \right\rangle\left\langle 1 \right|)\end{aligned}\] In matrix form with respect to the computational basis $\{\left| 0 \right\rangle, \left| 1 \right\rangle\}$, the Hadamard operator is represented as: \[H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\]

For $n$ qubits, applying the Hadamard gate to each qubit independently is represented by the $n$-fold tensor product $H^{\otimes n}$. Its action on the state $\left| 0 \right\rangle^{\otimes n} = \left| 00\cdots 0 \right\rangle$ is: \[H^{\otimes n} \left| 0 \right\rangle^{\otimes n} = H\left| 0 \right\rangle \otimes H\left| 0 \right\rangle \otimes \cdots \otimes H\left| 0 \right\rangle = \left( \frac{\left| 0 \right\rangle + \left| 1 \right\rangle}{\sqrt{2}} \right) \otimes \cdots \otimes \left( \frac{\left| 0 \right\rangle + \left| 1 \right\rangle}{\sqrt{2}} \right) = \frac{1}{2^{n/2}} \sum_{x \in \{0,1\}^n} \left| x \right\rangle\] where $x = (x_1, x_2, \dots, x_n)$ is an $n$-bit string, and $\left| x \right\rangle = \left| x_1 \right\rangle \otimes \left| x_2 \right\rangle \otimes \cdots \otimes \left| x_n \right\rangle$.

The Hadamard operator can also be expressed in a compact form using a sum over computational basis states. For a single qubit Hadamard gate, we can write: \[H = \frac{1}{\sqrt{2}} \sum_{x \in \{0,1\}} \sum_{y \in \{0,1\}} (-1)^{xy} \left| x \right\rangle\left\langle y \right|\] where $xy$ is the product of $x$ and $y$. Let’s verify this formula by expanding the summation:

For $x=0, y=0$: $(-1)^{0 \cdot 0} \left| 0 \right\rangle\left\langle 0 \right| = \left| 0 \right\rangle\left\langle 0 \right|$
For $x=0, y=1$: $(-1)^{0 \cdot 1} \left| 0 \right\rangle\left\langle 1 \right| = \left| 0 \right\rangle\left\langle 1 \right|$
For $x=1, y=0$: $(-1)^{1 \cdot 0} \left| 1 \right\rangle\left\langle 0 \right| = \left| 1 \right\rangle\left\langle 0 \right|$
For $x=1, y=1$: $(-1)^{1 \cdot 1} \left| 1 \right\rangle\left\langle 1 \right| = -\left| 1 \right\rangle\left\langle 1 \right|$

Summing these terms and multiplying by $\frac{1}{\sqrt{2}}$, we get $H = \frac{1}{\sqrt{2}} (\left| 0 \right\rangle\left\langle 0 \right| + \left| 0 \right\rangle\left\langle 1 \right| + \left| 1 \right\rangle\left\langle 0 \right| - \left| 1 \right\rangle\left\langle 1 \right|)$, which matches our earlier derivation.

For $H^{\otimes n}$, the operator can be written as: \[H^{\otimes n} = \frac{1}{2^{n/2}} \sum_{x \in \{0,1\}^n} \sum_{y \in \{0,1\}^n} (-1)^{x \cdot y} \left| x \right\rangle\left\langle y \right|\] where $x \cdot y = \sum_{i=1}^n x_i y_i$ is the dot product of the binary vectors $x = (x_1, \dots, x_n)$ and $y = (y_1, \dots, y_n)$. This compact form is particularly useful in analyzing quantum algorithms like the Quantum Fourier Transform.

Introduction to Quantum Mechanics

Motivation for Quantum Mechanics

Limitations of Classical Physics

By the late 19th and early 20th centuries, classical physics, encompassing Newtonian mechanics and Maxwell’s electromagnetism, encountered fundamental limitations in explaining phenomena at the atomic and molecular level. Several experimental observations at this scale could not be reconciled with classical theories, necessitating the development of quantum mechanics. Classical theories, while successful at macroscopic scales, failed to explain phenomena observed in microscopic systems.

Blackbody Radiation and Planck’s Quantization

Blackbody radiation is the electromagnetic radiation emitted by an object in thermal equilibrium. Classical theory, using the Rayleigh-Jeans law, predicted that the energy density of blackbody radiation increases with the square of the frequency. This led to the ultraviolet catastrophe, an unphysical divergence of energy at high frequencies, contradicting experimental observations.

In 1900, Max Planck resolved this issue by introducing energy quantization. He postulated that energy exchange between matter and radiation of frequency $\nu$ occurs in discrete packets, or quanta, with energy $E = h\nu$, where $h$ is Planck’s constant. This quantization hypothesis led to Planck’s law, which accurately describes the blackbody spectrum and resolves the ultraviolet catastrophe by limiting high-frequency modes.

Photoelectric Effect and Photons

The photoelectric effect is the emission of electrons from a material when light shines on it. Classical physics predicted that the kinetic energy of emitted electrons should depend on the intensity of the incident light. However, experiments showed that the kinetic energy depends on the frequency of light, and there is a threshold frequency below which no electrons are emitted, regardless of intensity.

In 1905, Albert Einstein explained the photoelectric effect by proposing that light itself is quantized into particles called photons, each carrying energy $E = h\nu$. Electron emission occurs when a photon’s energy exceeds the material’s work function $\Phi$. The kinetic energy of emitted electrons is $E_k = h\nu - \Phi$. This explanation established the particle nature of light and earned Einstein the Nobel Prize in Physics, highlighting the quantum nature of light interaction with matter.

Atomic Spectra and Discrete Energy Levels

When elements are excited, they emit light at discrete frequencies, producing atomic spectra with sharp spectral lines. Classical physics could not explain these discrete spectra, as it predicted continuous emission. The discrete nature of atomic spectra suggested that electrons in atoms can only exist at specific, quantized energy levels. Transitions between these levels result in the emission or absorption of photons at discrete frequencies, corresponding to the observed spectral lines. This quantization of energy levels is a cornerstone of atomic structure and quantum mechanics.

Early Quantum Theory

Planck’s Energy Quanta

Planck’s hypothesis of energy quantization was the first step towards quantum theory. It suggested that energy exchange between radiation and matter is not continuous but discrete, occurring in multiples of $h\nu$. This was a radical departure from classical physics and laid the foundation for quantum mechanics.

Bohr’s Atomic Model

In 1913, Niels Bohr proposed a model for the hydrogen atom that incorporated quantization to explain its atomic spectrum. Bohr’s postulates were:

Electrons orbit the nucleus in specific stationary states without radiating energy, contrary to classical electromagnetism which predicts radiating charges.
Transitions between stationary states occur by absorbing or emitting a photon with energy equal to the energy difference between levels: $E = h\nu = E_2 - E_1$. This explains the discrete spectral lines.
The angular momentum of an electron in a stationary state is quantized as integer multiples of $\hbar = h/2\pi$: $L = n\hbar$, where $n \in \{1, 2, 3, \dots\}$. This quantization condition restricts electron orbits to specific radii and energies.

Bohr’s model successfully predicted the hydrogen spectrum and introduced quantized angular momentum and discrete energy levels, although it was eventually superseded by more complete quantum mechanical models.

Formulation of Quantum Mechanics

Wave and Matrix Mechanics

In the mid-1920s, two independent formulations of quantum mechanics were developed:

Wave Mechanics, formulated by Erwin Schrödinger, describes quantum systems using wave functions that evolve according to the Schrödinger equation. This approach emphasizes the wave-like nature of particles and is mathematically formulated using differential equations.
Matrix Mechanics, developed by Werner Heisenberg, Max Born, and Pascual Jordan, uses matrices to represent physical quantities (observables) and focuses on transitions between energy levels. This approach emphasizes the particle-like nature and uses linear algebra as its mathematical framework.

Paul Dirac later demonstrated the mathematical equivalence of these two formulations, showing they are different representations of the same underlying quantum theory. They are unified within the more abstract Hilbert space formalism.

Hilbert Space Formalism

The modern, mathematically rigorous formulation of quantum mechanics is based on Hilbert spaces. In this framework:

Quantum states are represented by vectors in a complex Hilbert space, denoted as $\left| \psi \right\rangle$. These are state vectors or wave functions when represented in a continuous basis like position. Hilbert space provides the abstract vector space for quantum states.
Observables are represented by Hermitian operators actingon this Hilbert space. Hermitian operators ensure that the eigenvalues, representing measurement outcomes, are real.

Wave functions $\psi(x, t)$ in Schrödinger’s theory are components of abstract state vectors $\left| \psi \right\rangle$ when projected onto a specific basis, such as the position basis $\left| x \right\rangle$. The Hilbert space formalism provides a general and powerful mathematical language for quantum mechanics, applicable to both continuous and discrete quantum systems.

Fundamental Postulates of Quantum Mechanics

The postulates of quantum mechanics are the foundational rules that cannot be derived from other principles and are accepted as the basis for describing and predicting the behavior of quantum systems.

State Space Postulate

Postulate 1 (State Space): The state of any isolated quantum system is completely described by a vector $\left| \psi \right\rangle$ in a complex Hilbert space. This vector is known as the state vector, and it is normalized, i.e., $\left\langle \psi \middle| \psi \right\rangle = 1$. The Hilbert space provides the mathematical arena for all possible states of the system.

Observables Postulate

Postulate 2 (Observables): Every physically observable quantity (such as energy, momentum, position, angular momentum) of a quantum system is represented by a Hermitian operator $O$ acting on the Hilbert space of states. Hermitian operators are chosen because their eigenvalues are real, corresponding to physically measurable quantities.

Measurement Postulate

Postulate 3 (Measurement): The possible outcomes of measuring an observable $O$ are the eigenvalues $\{\lambda_i\}$ of the corresponding Hermitian operator $O$. When a measurement is performed, the result will always be one of these eigenvalues. If the system is in an eigenstate $\left| \psi_i \right\rangle$ of $O$ with eigenvalue $\lambda_i$ ($O\left| \psi_i \right\rangle = \lambda_i \left| \psi_i \right\rangle$), then a measurement of $O$ will definitely yield the value $\lambda_i$.

Probabilistic Interpretation Postulate

Postulate 4 (Probabilistic Interpretation): When measuring an observable $O$ on a system in an arbitrary state $\left| \psi \right\rangle$, the probability of obtaining an eigenvalue $\lambda_i$ as the measurement outcome is given by Born’s rule: $P(\lambda_i) = |\left\langle \psi_i \middle| \psi \right\rangle|^2$, where $\left| \psi_i \right\rangle$ is the normalized eigenvector corresponding to the eigenvalue $\lambda_i$. The expectation value (average value) of the observable $O$ in the state $\left| \psi \right\rangle$ is given by: \[\langle O \rangle = \left\langle \psi \middle| O\psi \right\rangle = \sum_i \lambda_i P(\lambda_i) = \sum_i \lambda_i |\left\langle \psi_i \middle| \psi \right\rangle|^2\] This postulate introduces the probabilistic nature of quantum measurements.

Superposition Postulate

Postulate 5 (Superposition): If $\left| \psi_1 \right\rangle$ and $\left| \psi_2 \right\rangle$ are valid states of a quantum system, then any linear superposition $\left| \psi \right\rangle = c_1\left| \psi_1 \right\rangle + c_2\left| \psi_2 \right\rangle$ (where $c_1, c_2 \in \mathbb{C}$) is also a valid state. This principle, known as the superposition principle, allows quantum systems to exist in a combination of multiple states simultaneously, a concept fundamentally different from classical physics. Superposition is crucial for quantum computing and quantum phenomena like interference.

Correspondence Principle and Schrödinger Equation

Quantum Operators from Classical Variables

The correspondence principle provides a conceptual bridge between classical and quantum mechanics. It suggests that in the limit of large quantum numbers or classical conditions, quantum mechanics should reproduce classical mechanics. It also serves as a heuristic guide for constructing quantum operators from classical variables. In one dimension, key correspondences are:

Position $x \rightarrow \hat{x} = x$ (position operator, represented as multiplication by $x$ in position space)
Momentum $p \rightarrow \hat{p} = -i\hbar \frac{\partial}{\partial x}$ (momentum operator, represented as a derivative operator in position space)
Energy $E \rightarrow \hat{H} = i\hbar \frac{\partial}{\partial t}$ (Hamiltonian operator, representing the total energy of the system and related to time evolution)

These rules are not strict derivations but rather guidelines to transition from classical descriptions to quantum operators.

Schrödinger Equation: Time Evolution of Quantum Systems

The Schrödinger equation is the fundamental equation of motion in non-relativistic quantum mechanics. It describes how the state of a quantum system evolves in time. Derived using the correspondence principle and by quantizing the classical Hamiltonian, for a particle of mass $m$ in a potential $V(x)$, the time-dependent Schrödinger equation is: \[i\hbar \frac{\partial}{\partial t} \left| \psi(t) \right\rangle = \hat{H} \left| \psi(t) \right\rangle = \left( \frac{\hat{p}^2}{2m} + V(\hat{x}) \right) \left| \psi(t) \right\rangle = \left( -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x) \right) \left| \psi(t) \right\rangle\] This linear, first-order differential equation in time governs the time evolution of quantum states. The Hamiltonian operator $\hat{H}$ represents the total energy of the system and dictates the system’s dynamics. Eigenvalues of the Hamiltonian correspond to the allowed energy levels of the system. The Schrödinger equation is central to understanding quantum dynamics and is used extensively in all areas of quantum mechanics, including quantum computing.

Conclusion

This lecture provided a foundational overview of linear algebra essential for quantum mechanics and introduced the fundamental principles of quantum mechanics. We reviewed matrix diagonalization, adjoint operators, normal and Hermitian operators, and tensor products as key linear algebra tools. Transitioning to quantum mechanics, we discussed the historical motivations arising from the limitations of classical physics in explaining blackbody radiation, the photoelectric effect, and atomic spectra. We then explored early quantum theories by Planck and Bohr, and outlined the core postulates of quantum mechanics: state space, observables, measurement, probabilistic interpretation, and superposition. Finally, we introduced the correspondence principle and the Schrödinger equation, establishing a link between classical and quantum mechanics. Key concepts emphasized include energy quantization, wave-particle duality, and the probabilistic nature of quantum measurements. These principles and mathematical tools are crucial for understanding quantum phenomena and form the basis for quantum computing. Future lectures will expand on these foundations to explore quantum dynamics, quantum information, and quantum computation in detail.

--- title: "Lecture Notes on Linear Algebra and Introduction to Quantum Mechanics" 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 reviews matrix diagonalization and explores linear algebra exercises relevant to quantum mechanics. It then transitions into an introduction to quantum mechanics, discussing its motivations, early developments, formulations, and fundamental postulates. The lecture concludes by connecting quantum and classical mechanics through the correspondence principle and introducing the Schrödinger equation. # Linear Algebra Exercises ## Example of a Non-Diagonalizable Matrix ::: tcolorbox We aim to demonstrate that the matrix $$M = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$$ cannot be diagonalized. To determine if $M$ is diagonalizable, we first find its eigenvalues by solving the characteristic equation $\det(M - \lambda I) = 0$, where $I$ is the $2 \times 2$ identity matrix and $\lambda$ represents the eigenvalues. $$\begin{aligned}\det(M - \lambda I) &= \det \left( \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} - \lambda \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \right) \\&= \det \begin{pmatrix} 1-\lambda & 0 \\ 1 & 1-\lambda \end{pmatrix} \\&= (1-\lambda)(1-\lambda) - (0)(1) \\&= (1-\lambda)^2\end{aligned}$$ Setting the determinant to zero, $(1-\lambda)^2 = 0$, we find a repeated eigenvalue $\lambda = 1$ with algebraic multiplicity 2. Next, we find the eigenvectors corresponding to $\lambda = 1$ by solving the homogeneous system $(M - \lambda I)v = 0$: $$\begin{pmatrix} 1-1 & 0 \\ 1 & 1-1 \end{pmatrix} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$$ $$\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$$ This matrix equation is equivalent to the system of linear equations: $$\begin{aligned}0\alpha + 0\beta &= 0 \\1\alpha + 0\beta &= 0\end{aligned}$$ From the second equation, we have $\alpha = 0$. The variable $\beta$ is a free variable, meaning it can be any complex number. Choosing $\beta = 1$, we obtain an eigenvector $v = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$. Thus, all eigenvectors are of the form $c \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ for any non-zero scalar $c$. The geometric multiplicity of the eigenvalue $\lambda = 1$ is the dimension of the eigenspace, which is 1 (since all eigenvectors are scalar multiples of each other). Since the geometric multiplicity (1) is less than the algebraic multiplicity (2), the matrix $M$ is not diagonalizable. ::: ## Standard Procedure for Eigenvalues and Eigenvectors {#subsec:eigenvalues_eigenvectors} ::::: tcolorbox :::: algorithm_env **Algorithm 1** (H). ::: algorithmic **Form the Characteristic Equation:** Construct the characteristic equation $\det(A - \lambda I) = 0$, where $I$ is the identity matrix of the same dimension as $A$, and $\lambda$ represents the eigenvalues. **Solve for Eigenvalues:** Solve the characteristic equation for $\lambda$. The roots $\lambda_1, \lambda_2, \dots, \lambda_n$ of this polynomial are the eigenvalues of $A$. **Determine Eigenvectors:** For each eigenvalue $\lambda_i$, solve the homogeneous system of linear equations $(A - \lambda_i I)v_i = 0$ to find the corresponding eigenvector(s) $v_i$. The set of all non-zero solutions $v_i$ forms the eigenspace associated with $\lambda_i$. ::: :::: ::::: ## Adjoint (Dagger) of an Operator ::::: tcolorbox This theorem derives the adjoint of an operator defined as the outer product of two vectors. ::: proposition **Proposition 1**. *For an operator $O = \left| U \right\rangle\left\langle W \right|$, its adjoint is $O^\dagger = \left| W \right\rangle\left\langle U \right|$.* ::: ::: proof *Proof.* To prove this, we use the definition of the adjoint operator in terms of its matrix elements. For any operators $A$ and $B$ and vectors $\left| \phi \right\rangle$ and $\left| \psi \right\rangle$, the adjoint is defined such that $\left\langle \phi \right| A^\dagger \left| \psi \right\rangle = (\left\langle \psi \right| A \left| \phi \right\rangle)^*$. Let $O = \left| U \right\rangle\left\langle W \right|$. Then for arbitrary vectors $\left| I \right\rangle$ and $\left| J \right\rangle$: $$\begin{aligned}\left\langle J \right| O^\dagger \left| I \right\rangle &= (\left\langle I \right| O \left| J \right\rangle)^* \\&= (\left\langle I \right| (\left| U \right\rangle\left\langle W \right|) \left| J \right\rangle)^* \\&= (\left\langle I \middle| U \right\rangle \left\langle W \middle| J \right\rangle)^* \\&= \left\langle I \middle| U \right\rangle^* \left\langle W \middle| J \right\rangle^* \\&= \left\langle U \middle| I \right\rangle \left\langle J \middle| W \right\rangle \\&= \left\langle J \middle| W \right\rangle \left\langle U \middle| I \right\rangle \\&= \left\langle J \right| (\left| W \right\rangle\left\langle U \right|) \left| I \right\rangle\end{aligned}$$ Since this equality holds for all vectors $\left| I \right\rangle$ and $\left| J \right\rangle$, we conclude that the operator $O^\dagger$ is indeed $\left| W \right\rangle\left\langle U \right|$. ◻ ::: ::::: ## Normal and Hermitian Operators ### Definitions ::: tcolorbox An operator $A$ is **Hermitian** if it is equal to its adjoint, i.e., $A = A^\dagger$. Hermitian operators represent physical observables in quantum mechanics. ::: ::: tcolorbox An operator $A$ is **normal** if it commutes with its adjoint, i.e., $AA^\dagger = A^\dagger A$. Normal operators are important because they are diagonalizable by a unitary transformation. ::: ### Hermitian Condition for Normal Operators ::::: tcolorbox This theorem states the necessary and sufficient condition for a normal matrix to be Hermitian based on its eigenvalues. ::: theorem **Theorem 2**. *A normal matrix $A$ is Hermitian if and only if all its eigenvalues are real.* ::: ::: proof *Proof.* Let $A$ be a normal matrix. By the spectral theorem for normal operators, there exists a unitary matrix $U$ such that $A = UDU^\dagger$, where $D$ is a diagonal matrix whose diagonal entries are the eigenvalues of $A$. The adjoint of $A$ is $A^\dagger = (UDU^\dagger)^\dagger = (U^\dagger)^\dagger D^\dagger U^\dagger = UD^\dagger U^\dagger$. ($\Rightarrow$) Assume $A$ is Hermitian, so $A = A^\dagger$. Then $UDU^\dagger = UD^\dagger U^\dagger$. Multiplying by $U^\dagger$ on the left and $U$ on the right, we obtain $D = D^\dagger$. For a diagonal matrix, the adjoint is obtained by taking the complex conjugate of each diagonal entry. Thus, $D = D^\dagger$ implies that all diagonal entries of $D$, which are the eigenvalues of $A$, must be real. ($\Leftarrow$) Assume all eigenvalues of $A$ are real. Then the diagonal matrix $D$ has real entries, so $D = D^\dagger$. Therefore, $A^\dagger = UD^\dagger U^\dagger = UDU^\dagger = A$. Hence, $A$ is Hermitian. ◻ ::: ::::: ### Spectral Resolution A normal operator $A$ can be decomposed using its spectral resolution as: $$A = \sum_{k} \lambda_k P_k$$ where $\lambda_k$ are the eigenvalues of $A$, and $P_k$ are the orthogonal projectors onto the eigenspaces corresponding to $\lambda_k$. For a Hermitian operator, the eigenvalues $\lambda_k$ are guaranteed to be real, and the eigenvectors corresponding to distinct eigenvalues are orthogonal. ## Tensor Product Operations ### Tensor Product of State Vectors Given a single-qubit state $\left| \psi \right\rangle = \frac{1}{\sqrt{2}} (\left| 0 \right\rangle + \left| 1 \right\rangle)$, the tensor product of two such states is: $$\begin{aligned}\left| \psi \right\rangle \otimes \left| \psi \right\rangle &= \left( \frac{1}{\sqrt{2}} (\left| 0 \right\rangle + \left| 1 \right\rangle) \right) \otimes \left( \frac{1}{\sqrt{2}} (\left| 0 \right\rangle + \left| 1 \right\rangle) \right) \\&= \frac{1}{2} (\left| 0 \right\rangle \otimes \left| 0 \right\rangle + \left| 0 \right\rangle \otimes \left| 1 \right\rangle + \left| 1 \right\rangle \otimes \left| 0 \right\rangle + \left| 1 \right\rangle \otimes \left| 1 \right\rangle) \\&= \frac{1}{2} (\left| 00 \right\rangle + \left| 01 \right\rangle + \left| 10 \right\rangle + \left| 11 \right\rangle)\end{aligned}$$ For $n$ qubits, the $n$-fold tensor product is: $$\left| \psi \right\rangle^{\otimes n} = \left( \frac{1}{\sqrt{2}} (\left| 0 \right\rangle + \left| 1 \right\rangle) \right)^{\otimes n} = \frac{1}{2^{n/2}} \sum_{x_1 \in \{0,1\}} \cdots \sum_{x_n \in \{0,1\}} \left| x_1 \cdots x_n \right\rangle = \frac{1}{2^{n/2}} \sum_{x \in \{0,1\}^n} \left| x \right\rangle$$ This represents an equal superposition over all $2^n$ computational basis states, which is crucial in quantum algorithms. ### Tensor Product of Operators If an operator $A$ acts on a vector space $V_1$ and an operator $B$ acts on a vector space $V_2$, their tensor product $A \otimes B$ acts on the tensor product space $V_1 \otimes V_2$. In matrix form, if $A$ is an $m \times m$ matrix and $B$ is an $n \times n$ matrix, the tensor product $A \otimes B$ is an $mn \times mn$ matrix. If $A = (a_{ij})$ and $B = (b_{kl})$, the matrix for $A \otimes B$ can be constructed in blocks. For example, if $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$, then $$A \otimes B = \begin{pmatrix} a_{11}B & a_{12}B \\ a_{21}B & a_{22}B \end{pmatrix}$$ where each block $a_{ij}B$ is a matrix obtained by multiplying each entry of $B$ by the scalar $a_{ij}$. ### Transpose and Adjoint Properties of Tensor Products ::::: tcolorbox This theorem demonstrates that the transpose operation distributes over the tensor product. ::: proposition **Proposition 3**. *The transpose of a tensor product is the tensor product of the transposes: $(A \otimes B)^T = A^T \otimes B^T$.* ::: ::: proof *Proof.* Let $A$ be an $m \times m$ matrix and $B$ be an $n \times n$ matrix. The $((i, k), (j, l))$-th element of $A \otimes B$ is given by $(A \otimes B)_{(i, k), (j, l)} = A_{ij} B_{kl}$. The transpose operation swaps rows and columns. Thus, the $((j, l), (i, k))$-th element of $(A \otimes B)^T$ is $((A \otimes B)^T)_{(j, l), (i, k)} = (A \otimes B)_{(i, k), (j, l)} = A_{ij} B_{kl}$. On the other hand, the $((j, l), (i, k))$-th element of $A^T \otimes B^T$ is $(A^T \otimes B^T)_{(j, l), (i, k)} = (A^T)_{ji} (B^T)_{lk} = A_{ij} B_{kl}$. Since the elements are equal, we conclude that $(A \otimes B)^T = A^T \otimes B^T$. ◻ ::: ::::: ::::: tcolorbox This theorem demonstrates that the adjoint operation distributes over the tensor product. ::: proposition **Proposition 4**. *The adjoint of a tensor product is the tensor product of the adjoints: $(A \otimes B)^\dagger = A^\dagger \otimes B^\dagger$.* ::: ::: proof *Proof.* Using the property that for any matrix $M$, $(M^\dagger) = (M^T)^* = (M^*)^T$, we have: $$(A \otimes B)^\dagger = ((A \otimes B)^T)^* = (A^T \otimes B^T)^* = (A^T)^* \otimes (B^T)^* = A^\dagger \otimes B^\dagger$$ ◻ ::: ::::: ::::: tcolorbox This theorem shows that the transpose of a product of operators is the product of transposes in reverse order. ::: proposition **Proposition 5**. *The transpose of a product of operators is the product of transposes in reverse order: $(AB)^T = B^T A^T$.* ::: ::: proof *Proof.* Let $A$ and $B$ be matrices such that their product $AB$ is defined. The $(i, j)$-th element of $AB$ is $(AB)_{ij} = \sum_k A_{ik} B_{kj}$. The transpose operation swaps indices, so the $(j, i)$-th element of $(AB)^T$ is $((AB)^T)_{ji} = (AB)_{ij} = \sum_k A_{ik} B_{kj}$. Now consider the product $B^T A^T$. Its $(j, i)$-th element is $(B^T A^T)_{ji} = \sum_k (B^T)_{jk} (A^T)_{ki} = \sum_k B_{kj} A_{ik} = \sum_k A_{ik} B_{kj}$. Since $((AB)^T)_{ji} = (B^T A^T)_{ji}$ for all $i, j$, we conclude that $(AB)^T = B^T A^T$. ◻ ::: ::::: ::::: tcolorbox This theorem shows that the adjoint of a product of operators is the product of adjoints in reverse order. ::: proposition **Proposition 6**. *The adjoint of a product of operators is the product of the adjoints in reverse order: $(AB)^\dagger = B^\dagger A^\dagger$.* ::: ::: proof *Proof.* Using the property that $(M^\dagger) = (M^T)^* = (M^*)^T$ and the transpose property of a product of operators, we have: $$(AB)^\dagger = ((AB)^T)^* = (B^T A^T)^* = (B^T)^* (A^T)^* = B^\dagger A^\dagger$$ ◻ ::: ::::: ## Representation of the Hadamard Operator The Hadamard operator $H$ is a fundamental single-qubit gate in quantum computing, defined by its action on the computational basis states $\left| 0 \right\rangle$ and $\left| 1 \right\rangle$: $$\begin{aligned}H\left| 0 \right\rangle &= \frac{1}{\sqrt{2}} (\left| 0 \right\rangle + \left| 1 \right\rangle) \\H\left| 1 \right\rangle &= \frac{1}{\sqrt{2}} (\left| 0 \right\rangle - \left| 1 \right\rangle)\end{aligned}$$ We can express the Hadamard operator using projectors onto the computational basis states: $$\begin{aligned}H &= H \left| 0 \right\rangle\left\langle 0 \right| + H \left| 1 \right\rangle\left\langle 1 \right| \\&= \frac{1}{\sqrt{2}} (\left| 0 \right\rangle + \left| 1 \right\rangle)\left\langle 0 \right| + \frac{1}{\sqrt{2}} (\left| 0 \right\rangle - \left| 1 \right\rangle)\left\langle 1 \right| \\&= \frac{1}{\sqrt{2}} (\left| 0 \right\rangle\left\langle 0 \right| + \left| 1 \right\rangle\left\langle 0 \right| + \left| 0 \right\rangle\left\langle 1 \right| - \left| 1 \right\rangle\left\langle 1 \right|) \\&= \frac{1}{\sqrt{2}} (\left| 0 \right\rangle\left\langle 0 \right| + \left| 0 \right\rangle\left\langle 1 \right| + \left| 1 \right\rangle\left\langle 0 \right| - \left| 1 \right\rangle\left\langle 1 \right|)\end{aligned}$$ In matrix form with respect to the computational basis $\{\left| 0 \right\rangle, \left| 1 \right\rangle\}$, the Hadamard operator is represented as: $$H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$$ For $n$ qubits, applying the Hadamard gate to each qubit independently is represented by the $n$-fold tensor product $H^{\otimes n}$. Its action on the state $\left| 0 \right\rangle^{\otimes n} = \left| 00\cdots 0 \right\rangle$ is: $$H^{\otimes n} \left| 0 \right\rangle^{\otimes n} = H\left| 0 \right\rangle \otimes H\left| 0 \right\rangle \otimes \cdots \otimes H\left| 0 \right\rangle = \left( \frac{\left| 0 \right\rangle + \left| 1 \right\rangle}{\sqrt{2}} \right) \otimes \cdots \otimes \left( \frac{\left| 0 \right\rangle + \left| 1 \right\rangle}{\sqrt{2}} \right) = \frac{1}{2^{n/2}} \sum_{x \in \{0,1\}^n} \left| x \right\rangle$$ where $x = (x_1, x_2, \dots, x_n)$ is an $n$-bit string, and $\left| x \right\rangle = \left| x_1 \right\rangle \otimes \left| x_2 \right\rangle \otimes \cdots \otimes \left| x_n \right\rangle$. The Hadamard operator can also be expressed in a compact form using a sum over computational basis states. For a single qubit Hadamard gate, we can write: $$H = \frac{1}{\sqrt{2}} \sum_{x \in \{0,1\}} \sum_{y \in \{0,1\}} (-1)^{xy} \left| x \right\rangle\left\langle y \right|$$ where $xy$ is the product of $x$ and $y$. Let's verify this formula by expanding the summation: - For $x=0, y=0$: $(-1)^{0 \cdot 0} \left| 0 \right\rangle\left\langle 0 \right| = \left| 0 \right\rangle\left\langle 0 \right|$ - For $x=0, y=1$: $(-1)^{0 \cdot 1} \left| 0 \right\rangle\left\langle 1 \right| = \left| 0 \right\rangle\left\langle 1 \right|$ - For $x=1, y=0$: $(-1)^{1 \cdot 0} \left| 1 \right\rangle\left\langle 0 \right| = \left| 1 \right\rangle\left\langle 0 \right|$ - For $x=1, y=1$: $(-1)^{1 \cdot 1} \left| 1 \right\rangle\left\langle 1 \right| = -\left| 1 \right\rangle\left\langle 1 \right|$ Summing these terms and multiplying by $\frac{1}{\sqrt{2}}$, we get $H = \frac{1}{\sqrt{2}} (\left| 0 \right\rangle\left\langle 0 \right| + \left| 0 \right\rangle\left\langle 1 \right| + \left| 1 \right\rangle\left\langle 0 \right| - \left| 1 \right\rangle\left\langle 1 \right|)$, which matches our earlier derivation. For $H^{\otimes n}$, the operator can be written as: $$H^{\otimes n} = \frac{1}{2^{n/2}} \sum_{x \in \{0,1\}^n} \sum_{y \in \{0,1\}^n} (-1)^{x \cdot y} \left| x \right\rangle\left\langle y \right|$$ where $x \cdot y = \sum_{i=1}^n x_i y_i$ is the dot product of the binary vectors $x = (x_1, \dots, x_n)$ and $y = (y_1, \dots, y_n)$. This compact form is particularly useful in analyzing quantum algorithms like the Quantum Fourier Transform. # Introduction to Quantum Mechanics ## Motivation for Quantum Mechanics ### Limitations of Classical Physics By the late 19th and early 20th centuries, classical physics, encompassing Newtonian mechanics and Maxwell's electromagnetism, encountered fundamental limitations in explaining phenomena at the atomic and molecular level. Several experimental observations at this scale could not be reconciled with classical theories, necessitating the development of quantum mechanics. Classical theories, while successful at macroscopic scales, failed to explain phenomena observed in microscopic systems. ### Blackbody Radiation and Planck's Quantization ::: tcolorbox **Blackbody radiation** is the electromagnetic radiation emitted by an object in thermal equilibrium. Classical theory, using the Rayleigh-Jeans law, predicted that the energy density of blackbody radiation increases with the square of the frequency. This led to the **ultraviolet catastrophe**, an unphysical divergence of energy at high frequencies, contradicting experimental observations. ::: In 1900, Max Planck resolved this issue by introducing **energy quantization**. He postulated that energy exchange between matter and radiation of frequency $\nu$ occurs in discrete packets, or quanta, with energy $E = h\nu$, where $h$ is Planck's constant. This quantization hypothesis led to Planck's law, which accurately describes the blackbody spectrum and resolves the ultraviolet catastrophe by limiting high-frequency modes. ### Photoelectric Effect and Photons ::: tcolorbox The **photoelectric effect** is the emission of electrons from a material when light shines on it. Classical physics predicted that the kinetic energy of emitted electrons should depend on the intensity of the incident light. However, experiments showed that the kinetic energy depends on the *frequency* of light, and there is a threshold frequency below which no electrons are emitted, regardless of intensity. ::: In 1905, Albert Einstein explained the photoelectric effect by proposing that light itself is quantized into particles called **photons**, each carrying energy $E = h\nu$. Electron emission occurs when a photon's energy exceeds the material's work function $\Phi$. The kinetic energy of emitted electrons is $E_k = h\nu - \Phi$. This explanation established the particle nature of light and earned Einstein the Nobel Prize in Physics, highlighting the quantum nature of light interaction with matter. ### Atomic Spectra and Discrete Energy Levels ::: tcolorbox When elements are excited, they emit light at discrete frequencies, producing **atomic spectra** with sharp spectral lines. Classical physics could not explain these discrete spectra, as it predicted continuous emission. The discrete nature of atomic spectra suggested that electrons in atoms can only exist at specific, quantized energy levels. Transitions between these levels result in the emission or absorption of photons at discrete frequencies, corresponding to the observed spectral lines. This quantization of energy levels is a cornerstone of atomic structure and quantum mechanics. ::: ## Early Quantum Theory ### Planck's Energy Quanta ::: tcolorbox Planck's hypothesis of energy quantization was the first step towards quantum theory. It suggested that energy exchange between radiation and matter is not continuous but discrete, occurring in multiples of $h\nu$. This was a radical departure from classical physics and laid the foundation for quantum mechanics. ::: ### Bohr's Atomic Model ::: tcolorbox In 1913, Niels Bohr proposed a model for the hydrogen atom that incorporated quantization to explain its atomic spectrum. Bohr's postulates were: 1. Electrons orbit the nucleus in specific stationary states without radiating energy, contrary to classical electromagnetism which predicts radiating charges. 2. Transitions between stationary states occur by absorbing or emitting a photon with energy equal to the energy difference between levels: $E = h\nu = E_2 - E_1$. This explains the discrete spectral lines. 3. The angular momentum of an electron in a stationary state is quantized as integer multiples of $\hbar = h/2\pi$: $L = n\hbar$, where $n \in \{1, 2, 3, \dots\}$. This quantization condition restricts electron orbits to specific radii and energies. Bohr's model successfully predicted the hydrogen spectrum and introduced quantized angular momentum and discrete energy levels, although it was eventually superseded by more complete quantum mechanical models. ::: ## Formulation of Quantum Mechanics ### Wave and Matrix Mechanics ::: tcolorbox In the mid-1920s, two independent formulations of quantum mechanics were developed: - **Wave Mechanics**, formulated by Erwin Schrödinger, describes quantum systems using wave functions that evolve according to the Schrödinger equation. This approach emphasizes the wave-like nature of particles and is mathematically formulated using differential equations. - **Matrix Mechanics**, developed by Werner Heisenberg, Max Born, and Pascual Jordan, uses matrices to represent physical quantities (observables) and focuses on transitions between energy levels. This approach emphasizes the particle-like nature and uses linear algebra as its mathematical framework. Paul Dirac later demonstrated the mathematical equivalence of these two formulations, showing they are different representations of the same underlying quantum theory. They are unified within the more abstract Hilbert space formalism. ::: ### Hilbert Space Formalism ::: tcolorbox The modern, mathematically rigorous formulation of quantum mechanics is based on **Hilbert spaces**. In this framework: - **Quantum states** are represented by vectors in a complex Hilbert space, denoted as $\left| \psi \right\rangle$. These are state vectors or wave functions when represented in a continuous basis like position. Hilbert space provides the abstract vector space for quantum states. - **Observables** are represented by **Hermitian operators** actingon this Hilbert space. Hermitian operators ensure that the eigenvalues, representing measurement outcomes, are real. Wave functions $\psi(x, t)$ in Schrödinger's theory are components of abstract state vectors $\left| \psi \right\rangle$ when projected onto a specific basis, such as the position basis $\left| x \right\rangle$. The Hilbert space formalism provides a general and powerful mathematical language for quantum mechanics, applicable to both continuous and discrete quantum systems. ::: ## Fundamental Postulates of Quantum Mechanics The postulates of quantum mechanics are the foundational rules that cannot be derived from other principles and are accepted as the basis for describing and predicting the behavior of quantum systems. ### State Space Postulate ::: tcolorbox **Postulate 1 (State Space):** The state of any isolated quantum system is completely described by a vector $\left| \psi \right\rangle$ in a complex Hilbert space. This vector is known as the state vector, and it is normalized, i.e., $\left\langle \psi \middle| \psi \right\rangle = 1$. The Hilbert space provides the mathematical arena for all possible states of the system. ::: ### Observables Postulate ::: tcolorbox **Postulate 2 (Observables):** Every physically observable quantity (such as energy, momentum, position, angular momentum) of a quantum system is represented by a Hermitian operator $O$ acting on the Hilbert space of states. Hermitian operators are chosen because their eigenvalues are real, corresponding to physically measurable quantities. ::: ### Measurement Postulate ::: tcolorbox **Postulate 3 (Measurement):** The possible outcomes of measuring an observable $O$ are the eigenvalues $\{\lambda_i\}$ of the corresponding Hermitian operator $O$. When a measurement is performed, the result will always be one of these eigenvalues. If the system is in an eigenstate $\left| \psi_i \right\rangle$ of $O$ with eigenvalue $\lambda_i$ ($O\left| \psi_i \right\rangle = \lambda_i \left| \psi_i \right\rangle$), then a measurement of $O$ will definitely yield the value $\lambda_i$. ::: ### Probabilistic Interpretation Postulate ::: tcolorbox **Postulate 4 (Probabilistic Interpretation):** When measuring an observable $O$ on a system in an arbitrary state $\left| \psi \right\rangle$, the probability of obtaining an eigenvalue $\lambda_i$ as the measurement outcome is given by Born's rule: $P(\lambda_i) = |\left\langle \psi_i \middle| \psi \right\rangle|^2$, where $\left| \psi_i \right\rangle$ is the normalized eigenvector corresponding to the eigenvalue $\lambda_i$. The **expectation value** (average value) of the observable $O$ in the state $\left| \psi \right\rangle$ is given by: $$\langle O \rangle = \left\langle \psi \middle| O\psi \right\rangle = \sum_i \lambda_i P(\lambda_i) = \sum_i \lambda_i |\left\langle \psi_i \middle| \psi \right\rangle|^2$$ This postulate introduces the probabilistic nature of quantum measurements. ::: ### Superposition Postulate ::: tcolorbox **Postulate 5 (Superposition):** If $\left| \psi_1 \right\rangle$ and $\left| \psi_2 \right\rangle$ are valid states of a quantum system, then any linear superposition $\left| \psi \right\rangle = c_1\left| \psi_1 \right\rangle + c_2\left| \psi_2 \right\rangle$ (where $c_1, c_2 \in \mathbb{C}$) is also a valid state. This principle, known as the superposition principle, allows quantum systems to exist in a combination of multiple states simultaneously, a concept fundamentally different from classical physics. Superposition is crucial for quantum computing and quantum phenomena like interference. ::: ## Correspondence Principle and Schrödinger Equation ### Quantum Operators from Classical Variables ::: tcolorbox The **correspondence principle** provides a conceptual bridge between classical and quantum mechanics. It suggests that in the limit of large quantum numbers or classical conditions, quantum mechanics should reproduce classical mechanics. It also serves as a heuristic guide for constructing quantum operators from classical variables. In one dimension, key correspondences are: - Position $x \rightarrow \hat{x} = x$ (position operator, represented as multiplication by $x$ in position space) - Momentum $p \rightarrow \hat{p} = -i\hbar \frac{\partial}{\partial x}$ (momentum operator, represented as a derivative operator in position space) - Energy $E \rightarrow \hat{H} = i\hbar \frac{\partial}{\partial t}$ (Hamiltonian operator, representing the total energy of the system and related to time evolution) These rules are not strict derivations but rather guidelines to transition from classical descriptions to quantum operators. ::: ### Schrödinger Equation: Time Evolution of Quantum Systems ::: tcolorbox The **Schrödinger equation** is the fundamental equation of motion in non-relativistic quantum mechanics. It describes how the state of a quantum system evolves in time. Derived using the correspondence principle and by quantizing the classical Hamiltonian, for a particle of mass $m$ in a potential $V(x)$, the time-dependent Schrödinger equation is: $$i\hbar \frac{\partial}{\partial t} \left| \psi(t) \right\rangle = \hat{H} \left| \psi(t) \right\rangle = \left( \frac{\hat{p}^2}{2m} + V(\hat{x}) \right) \left| \psi(t) \right\rangle = \left( -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x) \right) \left| \psi(t) \right\rangle$$ This linear, first-order differential equation in time governs the time evolution of quantum states. The Hamiltonian operator $\hat{H}$ represents the total energy of the system and dictates the system's dynamics. Eigenvalues of the Hamiltonian correspond to the allowed energy levels of the system. The Schrödinger equation is central to understanding quantum dynamics and is used extensively in all areas of quantum mechanics, including quantum computing. ::: # Conclusion This lecture provided a foundational overview of linear algebra essential for quantum mechanics and introduced the fundamental principles of quantum mechanics. We reviewed matrix diagonalization, adjoint operators, normal and Hermitian operators, and tensor products as key linear algebra tools. Transitioning to quantum mechanics, we discussed the historical motivations arising from the limitations of classical physics in explaining blackbody radiation, the photoelectric effect, and atomic spectra. We then explored early quantum theories by Planck and Bohr, and outlined the core postulates of quantum mechanics: state space, observables, measurement, probabilistic interpretation, and superposition. Finally, we introduced the correspondence principle and the Schrödinger equation, establishing a link between classical and quantum mechanics. Key concepts emphasized include energy quantization, wave-particle duality, and the probabilistic nature of quantum measurements. These principles and mathematical tools are crucial for understanding quantum phenomena and form the basis for quantum computing. Future lectures will expand on these foundations to explore quantum dynamics, quantum information, and quantum computation in detail.