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Schrödinger Equation

The Formula

\[ i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r}, t) = \hat{H} \Psi(\mathbf{r}, t) \]

Where \(\hat{H}\) is the Hamiltonian operator. For a single particle in a potential \(V(\mathbf{r})\):

\[ \hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r}) \]

The time-independent form (for stationary states) is:

\[ \hat{H} \psi(\mathbf{r}) = E \psi(\mathbf{r}) \]

Or explicitly:

\[ -\frac{\hbar^2}{2m} \nabla^2 \psi(\mathbf{r}) + V(\mathbf{r}) \psi(\mathbf{r}) = E \psi(\mathbf{r}) \]

What It Means

The Schrödinger equation is the fundamental equation of quantum mechanics. It describes how the quantum state of a physical system changes over time.

The wave function \(\Psi(\mathbf{r}, t)\) contains all the information about a system. Its square modulus \(|\Psi|^2\) gives the probability density of finding a particle at position \(\mathbf{r}\) at time \(t\).

Think of it as the quantum equivalent of Newton's \(F = ma\): given the current state and the forces acting on a system, the Schrödinger equation predicts how it will evolve.

Why It Works — The Intuition

Classical mechanics describes particles as having definite positions and momenta. Quantum mechanics reveals that particles behave like waves — they don't have precise locations, but rather exist in a "cloud" of probabilities.

The key insight is that energy and momentum are related to wave properties:

  • Energy \(E\) is related to frequency: \(E = \hbar \omega\)
  • Momentum \(p\) is related to wavelength: \(p = \hbar k\)

The Schrödinger equation essentially says: "The rate of change of the wave function (left side) equals the total energy operator acting on the wave function (right side)."

The Hamiltonian \(\hat{H}\) represents the total energy: kinetic energy (\(-\frac{\hbar^2}{2m} \nabla^2\)) plus potential energy (\(V\)).

Derivation

Starting from Wave-Particle Duality

De Broglie proposed that particles have wave-like properties with:

\[ \lambda = \frac{h}{p}, \quad \omega = \frac{E}{\hbar} \]

A free particle can be described by a plane wave:

\[ \Psi(\mathbf{r}, t) = A e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)} \]

Energy Relations

For a non-relativistic particle, total energy is:

\[ E = \frac{p^2}{2m} + V \]

Using the operator correspondences:

\[ E \to i\hbar \frac{\partial}{\partial t}, \quad \mathbf{p} \to -i\hbar \nabla \]

Constructing the Equation

Apply these operators to the wave function. The energy operator gives:

\[ i\hbar \frac{\partial}{\partial t} \Psi = E \Psi \]

The kinetic energy operator gives:

\[ \frac{\mathbf{p}^2}{2m} \Psi = -\frac{\hbar^2}{2m} \nabla^2 \Psi \]

Therefore, the total energy equation becomes:

\[ i\hbar \frac{\partial}{\partial t} \Psi = \left(-\frac{\hbar^2}{2m} \nabla^2 + V\right) \Psi \]

This is the time-dependent Schrödinger equation.

Separation of Variables

For time-independent potentials, we can separate variables:

\[ \Psi(\mathbf{r}, t) = \psi(\mathbf{r}) e^{-iEt/\hbar} \]

Substituting into the time-dependent equation:

\[ i\hbar \left(-\frac{iE}{\hbar}\right) \psi e^{-iEt/\hbar} = \hat{H} \psi e^{-iEt/\hbar} \]

Simplifying:

\[ E \psi = \hat{H} \psi \]

This is the time-independent Schrödinger equation — an eigenvalue equation where \(E\) is the eigenvalue and \(\psi\) is the eigenfunction.

The Wave Function and Probability

The wave function \(\Psi\) is generally complex-valued. The Born rule states:

\[ P(\mathbf{r}, t) = |\Psi(\mathbf{r}, t)|^2 = \Psi^*(\mathbf{r}, t) \Psi(\mathbf{r}, t) \]

This is the probability density of finding the particle at position \(\mathbf{r}\) at time \(t\). The total probability must be 1:

\[ \int_{\text{all space}} |\Psi|^2 \, dV = 1 \]

This is the normalization condition.

Variables Explained

Symbol Name Description
\(\Psi(\mathbf{r}, t)\) Wave Function Complete quantum state of the system
\(\psi(\mathbf{r})\) Spatial Wave Function Time-independent part of \(\Psi\)
\(i\) Imaginary Unit \(\sqrt{-1}\), makes the equation complex
\(\hbar\) Reduced Planck Constant \(h/2\pi \approx 1.055 \times 10^{-34}\) J·s
\(\hat{H}\) Hamiltonian Operator Total energy operator
\(m\) Mass Mass of the particle
\(V(\mathbf{r})\) Potential Energy External potential at position \(\mathbf{r}\)
\(E\) Energy Eigenvalue Allowed energy levels
\(\nabla^2\) Laplacian \(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\)

Worked Example: Particle in a Box

Problem: A particle of mass \(m\) is confined to a 1D box of length \(L\), with \(V(x) = 0\) for \(0 < x < L\) and \(V = \infty\) elsewhere.

Step 1: Write the equation. Inside the box, \(V = 0\), so:

\[ -\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} = E \psi \]

Step 2: Solve the differential equation. Rearrange:

\[ \frac{d^2\psi}{dx^2} = -\frac{2mE}{\hbar^2} \psi = -k^2 \psi \]

where \(k = \sqrt{\frac{2mE}{\hbar^2}}\). The general solution is:

\[ \psi(x) = A \sin(kx) + B \cos(kx) \]

Step 3: Apply boundary conditions. The wave function must be zero at the walls:

  • \(\psi(0) = 0 \Rightarrow B = 0\)
  • \(\psi(L) = 0 \Rightarrow A \sin(kL) = 0\)

For non-trivial solutions, \(\sin(kL) = 0\), so:

\[ kL = n\pi, \quad n = 1, 2, 3, \ldots \]

Step 4: Find energy levels.

\[ k = \frac{n\pi}{L} = \sqrt{\frac{2mE}{\hbar^2}} \Rightarrow E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2} \]

Or in terms of \(h\):

\[ E_n = \frac{n^2 h^2}{8mL^2} \]

Step 5: Normalize the wave function.

\[ \psi_n(x) = A \sin\left(\frac{n\pi x}{L}\right) \]

Normalization: \(\int_0^L |\psi|^2 dx = 1\) gives \(A = \sqrt{\frac{2}{L}}\).

Result:

\[ \psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right), \quad E_n = \frac{n^2 h^2}{8mL^2} \]

The energy is quantized — only discrete values are allowed!

Common Mistakes

  • Thinking \(\Psi\) is physical: The wave function is a mathematical tool, not a physical wave. Only \(|\Psi|^2\) has direct physical meaning.
  • Forgetting the complex nature: The \(i\) in the equation is essential. Real-valued wave functions are special cases.
  • Confusing time-dependent and independent forms: Use the time-independent form only for stationary states (definite energy).
  • Ignoring boundary conditions: These determine the allowed energy levels and wave functions.
  • Thinking particles "orbit" like planets: Quantum particles don't have definite trajectories — they exist in probability distributions.