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Learning Objectives

By the end of this section, you will be able to:

Derive the time-independent Schrödinger equation from the time-dependent form using separation of variables
Understand the Hamiltonian operator as the quantum analog of total energy
Solve the eigenvalue problem for the infinite square well and interpret the quantized energy levels
Explain why energy quantization arises from boundary conditions
Connect eigenvalue problems in quantum mechanics to eigenvalue decomposition in machine learning
Implement numerical solutions of the TISE using finite differences

The Big Picture: Energy and Stationary States

"The equation that describes the stationary states of quantum systems is perhaps the most important equation in all of physics." — Richard Feynman

In the previous section, we introduced the time-dependent Schrödinger equation, which describes how quantum states evolve in time. But many of the most important questions in quantum mechanics concern stationary states — states whose probability distributions do not change with time. These are the states of definite energy.

The time-independent Schrödinger equation (TISE) is an eigenvalue problem: it asks which wave functions $\psi$ satisfy the property that applying the energy operator returns the same function multiplied by a constant. This constant is the energy $E$ .

Why This Matters

The time-independent Schrödinger equation gives us the allowed energy levels of quantum systems. This explains:

Atomic spectra: Why atoms emit and absorb light at specific wavelengths
Chemical bonds: Why atoms combine in specific ways to form molecules
Semiconductors: Why materials conduct, insulate, or exhibit intermediate behavior
Quantum computing: The discrete energy levels that encode qubits

Historical Context

When Schrödinger developed his equation in 1926, he was motivated by de Broglie's wave hypothesis and the success of Bohr's atomic model. He sought a wave equation that would naturally produce the quantized energy levels that Bohr had postulated. The time-independent equation emerged when he looked for wave functions that oscillate in time with a single frequency.

From Time-Dependent to Time-Independent

The time-dependent Schrödinger equation governs how a general quantum state evolves:

i\hbar \frac{\partial \Psi}{\partial t} = \hat{H}\Psi

where $\hat{H}$ is the Hamiltonian operator (total energy). We seek stationary states — solutions where the probability density $|\Psi|^2$ does not change with time. These can be found using separation of variables.

The Separation Ansatz

Assume the solution can be written as a product of spatial and temporal parts:

\Psi(x, t) = \psi(x) \cdot T(t)

Substituting into the time-dependent equation and dividing by $\psi T$ :

i\hbar \frac{1}{T}\frac{dT}{dt} = \frac{1}{\psi}\hat{H}\psi

The left side depends only on $t$ , the right side only on $x$ . For equality to hold for all $x$ and $t$ , both sides must equal a constant — call it $E$ .

The Two Separated Equations

Temporal Equation

i\hbar \frac{dT}{dt} = ET

Solution: $T(t) = e^{-iEt/\hbar}$

This is just oscillation at frequency $\omega = E/\hbar$ . The probability $|T|^2 = 1$ is constant.

Spatial Equation (TISE)

\hat{H}\psi = E\psi

This is the eigenvalue problem! Find which $\psi$ and $E$ satisfy this equation.

The full solution for a stationary state is therefore:

\Psi(x, t) = \psi(x) e^{-iEt/\hbar}

Probability is Stationary

The probability density is $|\Psi|^2 = |\psi|^2 |e^{-iEt/\hbar}|^2 = |\psi|^2$ , which is independent of time. This is why these states are called "stationary" — the probability distribution does not move or change.

The Hamiltonian Operator

The Hamiltonian operator $\hat{H}$ is the quantum mechanical operator corresponding to total energy. It consists of kinetic energy plus potential energy:

\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)

Kinetic energy operator + Potential energy operator

🔬The Hamiltonian Operator

Building the time-independent Schrödinger equation step by step

T̂

Kinetic Energy Operator

Classical kinetic energy is T = p²/2m. In quantum mechanics, we replace momentum p with the operator p̂ = -iℏ(d/dx).

T̂ = p̂²/2m = -ℏ²/(2m) · d²/dx²

Why d²/dx²?

The second derivative measures the curvature of the wave function. High curvature = short wavelength = high momentum = high kinetic energy. This connects to the de Broglie relation p = ℏk.

Eigenvalue Interpretation

When Ĥψ = Eψ, measuring energy always gives the value E with certainty. The state ψ is a "stationary state" — its probability density |ψ|² does not change with time.

Classical	→	Quantum Operator
x	→	x̂ = x (multiplication)
p	→	p̂ = -iℏ d/dx
T = p²/2m	→	T̂ = -ℏ²/(2m) d²/dx²
H = T + V	→	Ĥ = T̂ + V̂

Physical Interpretation of Each Term

Term	Classical Origin	Quantum Form	Physical Meaning
Kinetic	T = p²/2m	-ℏ²/(2m) d²/dx²	Curvature of ψ measures momentum
Potential	V(x)	V(x)ψ	Multiplication by potential function
Total	H = T + V	Ĥψ	Total energy of the quantum state

Why d²/dx²?

The second derivative measures the curvature of the wave function. Higher curvature means shorter wavelength, which by de Broglie means higher momentum. Since kinetic energy $T = p^2/2m$ , high curvature corresponds to high kinetic energy. The negative sign ensures positive kinetic energy when $\psi$ curves toward zero.

The Eigenvalue Problem

The time-independent Schrödinger equation is an eigenvalue equation:

The Time-Independent Schrödinger Equation

\hat{H}\psi = E\psi

Eigenfunction: $\psi$ — the wave function that satisfies this equation

Eigenvalue: $E$ — the energy of this state

Explicitly, for a one-dimensional system:

-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi(x) = E\psi(x)

Why "Eigenvalue"?

The German word "eigen" means "own" or "characteristic." The eigenvalue $E$ is the characteristic energy of the state $\psi$ . When you measure the energy of a particle in an eigenstate, you always get the eigenvalue $E$ with certainty.

The Eigenvalue Problem in Linear Algebra

If you discretize space (as we do numerically), the TISE becomes a matrix equation:

\mathbf{H}\vec{\psi} = E\vec{\psi}

This is exactly the eigenvalue problem from linear algebra! The Hamiltonian matrix $\mathbf{H}$ has eigenvalues (energies) and eigenvectors (discretized wave functions).

The Infinite Square Well: A Complete Example

The infinite square well (or "particle in a box") is the simplest non-trivial quantum system. It provides a complete, analytically solvable example that illustrates all the key concepts.

Setting Up the Problem

Consider a particle confined to a one-dimensional box of length $L$ :

V(x) = \begin{cases} 0 & 0 < x < L \\ \infty & \text{otherwise} \end{cases}

The infinite potential at the boundaries means the wave function must vanish there: $\psi(0) = \psi(L) = 0$ . Inside the box, the TISE becomes:

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

Solving the Differential Equation

This is a second-order linear ODE with constant coefficients. The general solution is:

\psi(x) = A\sin(kx) + B\cos(kx), \quad \text{where } k = \sqrt{2mE}/\hbar

Applying boundary conditions:

$\psi(0) = 0$ requires $B = 0$
$\psi(L) = 0$ requires $\sin(kL) = 0$ , so $kL = n\pi$ for $n = 1, 2, 3, \ldots$

The Eigenfunctions and Eigenvalues

Eigenfunctions

\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)

The factor $\sqrt{2/L}$ normalizes the wave function so that $\int_0^L |\psi_n|^2 dx = 1$ .

Eigenvalues (Energies)

E_n = \frac{n^2\pi^2\hbar^2}{2mL^2} = n^2 E_1

Energy is quantized — only these specific values are allowed. The ground state energy $E_1 > 0$ (zero-point energy).

⚛️Infinite Square Well: Eigenfunctions and Eigenvalues

Explore the stationary states ψ_n(x) and their energy eigenvalues E_n

Quantum Number n

Box Width L = 1.0 nm

Energy Levels

E₁= 1 E₁

E₂= 4 E₁

E₃= 9 E₁

Nodes: 0 (where ψ = 0)

The n-th eigenfunction has exactly (n-1) nodes inside the well.

Eigenfunction

ψ_n(x) = √(2/L) sin(nπx/L)

Eigenvalue

E_n = n²π²ℏ²/(2mL²)

Key Property

Energy grows as n² — larger gaps between higher levels

Physical Interpretation

The Wave Function

The wave function $\psi_n(x)$ is not directly observable, but its square $|\psi_n(x)|^2$ gives the probability density of finding the particle at position $x$ .

Ground State (n = 1)

The particle is most likely to be found in the center of the box. There are no nodes inside the box.

Excited States (n > 1)

Higher states have $n-1$ nodes — points where the probability of finding the particle is zero.

Nodes and Energy

There's a deep connection between nodes and energy:

More nodes = higher energy. The wave function must "wiggle" more to have more nodes, meaning shorter wavelength and higher kinetic energy.
Ground state has no internal nodes. It's the smoothest possible function satisfying the boundary conditions.
Higher excited states are orthogonal. The eigenfunctions with different $n$ are mutually orthogonal: $\int_0^L \psi_m \psi_n dx = 0$ for $m \neq n$ .

Zero-Point Energy

The lowest energy is $E_1 > 0$ , not zero! A confined particle cannot have zero kinetic energy. This is a direct consequence of the uncertainty principle: confining a particle to a box of size $L$ gives uncertainty in position $\Delta x \sim L$ , which implies uncertainty in momentum $\Delta p \gtrsim \hbar/L$ , and hence non-zero average kinetic energy.

Energy Level Quantization

One of the most profound consequences of quantum mechanics is that bound systems have discrete energy levels. The allowed energies are not continuous but come in specific, quantized values.

📊Energy Level Diagram

Quantized energy levels E_n = n²E₁ for a particle in an infinite square well

n=1

n=2

n=3

n=4

n=5

n=6

n=7

n=8

Click a level to see transitions

Box Width L = 1.0 nm

Energy scales as 1/L² — smaller boxes have larger energy gaps

Ground State Energy

E₁ = π²ℏ²/(2mL²)

Energy Gap

ΔE_n→m = (n² - m²)E₁

Why n²?

The n² dependence comes from fitting n half-wavelengths inside the box. Shorter wavelengths mean higher momentum (de Broglie: p = h/λ), and kinetic energy grows as p² = (h/λ)². Since λ = 2L/n, we get E ∝ n².

Why Does Quantization Occur?

Energy quantization arises from the boundary conditions. For the infinite square well:

The wave function must vanish at $x = 0$ and $x = L$
This requires fitting an integer number of half-wavelengths in the box: $\lambda_n = 2L/n$
By de Broglie, momentum $p_n = h/\lambda_n = nh/(2L)$
Kinetic energy $E_n = p_n^2/(2m) = n^2 h^2/(8mL^2)$

The boundary conditions act as a "selection rule" that allows only certain wavelengths (and hence energies) to exist.

Scaling with Size

The energy levels scale as $E_n \propto 1/L^2$ . This has important consequences:

System Size	Energy Spacing	Physical Implication
Atomic (L ~ 1 Å)	ΔE ~ eV	Visible light transitions
Nano (L ~ 10 nm)	ΔE ~ meV	Quantum dots, infrared
Macro (L ~ 1 m)	ΔE ~ 10⁻³⁷ eV	Effectively continuous

Classical Limit

As the box gets larger (or the particle gets heavier), the energy level spacing becomes so small that the spectrum appears continuous. This is how quantum mechanics reduces to classical mechanics for macroscopic objects — a key requirement for any valid quantum theory.

General Properties of Eigenfunctions

The eigenfunctions of the time-independent Schrödinger equation have several general properties that hold for any Hamiltonian:

Orthogonality

\int \psi_m^*(x)\psi_n(x)dx = \delta_{mn}

Eigenfunctions corresponding to different eigenvalues are orthogonal. This allows expansion of any state as a sum of energy eigenstates.

Completeness

\Psi(x) = \sum_n c_n\psi_n(x)

Any "reasonable" wave function can be expanded as a linear combination of the energy eigenfunctions.

Reality of Eigenvalues

E_n \in \mathbb{R}

The Hamiltonian is Hermitian, so all eigenvalues (energies) are real numbers. This is physically necessary — energy must be a real, measurable quantity.

Node Theorem

For a 1D potential, the n-th eigenfunction (ordered by energy) has exactly (n-1) nodes. The ground state has no nodes, first excited state has one node, etc.

Machine Learning Connections

The time-independent Schrödinger equation is an eigenvalue problem, and eigenvalue problems are ubiquitous in machine learning. Understanding this connection provides deep insight into both fields.

1. Principal Component Analysis (PCA)

PCA finds the eigenvalues and eigenvectors of the covariance matrix $\mathbf{C}$ :

\mathbf{C}\vec{v} = \lambda\vec{v}

Just as in quantum mechanics, the eigenvectors form an orthogonal basis (principal components), and the eigenvalues tell us the "energy" (variance) captured by each component.

2. Graph Neural Networks

GNNs often use the graph Laplacian $\mathbf{L} = \mathbf{D} - \mathbf{A}$ , which is the discrete analog of the Laplacian operator $\nabla^2$ in the Schrödinger equation. Spectral GNNs explicitly work with the eigenvectors of $\mathbf{L}$ .

3. Variational Quantum Eigensolver (VQE)

In quantum computing, VQE uses parameterized quantum circuits to find the ground state energy of molecular Hamiltonians — directly solving the TISE for complex molecules. This is one of the most promising near-term applications of quantum computers.

4. Neural Network Eigenvalue Problems

Physics-informed neural networks (PINNs) can be trained to solve eigenvalue problems by parameterizing $\psi(x)$ as a neural network and optimizing to satisfy $\hat{H}\psi = E\psi$ . This approach can handle complex potentials where analytical solutions don't exist.

Quantum Mechanics	Machine Learning Analog
Wave function ψ	Feature vector / latent representation
Hamiltonian Ĥ	Covariance matrix / Laplacian matrix
Energy eigenvalue E	Variance explained / spectral frequency
Ground state	Principal component 1 / dominant mode
Excited states	Higher principal components
Orthogonality	Decorrelated features

Python Implementation

Let's implement a numerical solver for the time-independent Schrödinger equation using finite differences. This method discretizes space onto a grid and converts the differential equation into a matrix eigenvalue problem.

Numerical Solution of the Time-Independent Schrödinger Equation

🐍solve_tise.py

Explanation(8)

Code(133)

3The TISE as a Matrix Eigenvalue Problem

The time-independent Schrödinger equation is a differential eigenvalue problem. Discretizing on a grid converts it to a matrix eigenvalue problem Hψ = Eψ, solvable with standard linear algebra.

22Spatial Grid

We create a grid excluding the boundary points where ψ = 0 (Dirichlet boundary conditions). This is equivalent to requiring the wave function to vanish at x = 0 and x = L.

31Finite Difference Approximation

The second derivative is approximated as (ψᵢ₊₁ - 2ψᵢ + ψᵢ₋₁)/dx². This three-point stencil gives a tridiagonal Hamiltonian matrix, which is computationally efficient to diagonalize.

40Tridiagonal Eigenvalue Solver

We use eigh_tridiagonal from SciPy, which is optimized for symmetric tridiagonal matrices. This is O(N²) instead of O(N³) for general matrices — a huge speedup for large N.

45Wavefunction Normalization

Physical wave functions must be normalized: ∫|ψ|²dx = 1. This ensures the total probability of finding the particle somewhere is exactly 1.

52Exact Solution for Comparison

For the infinite square well, we have analytical solutions: ψₙ(x) = √(2/L)sin(nπx/L) and Eₙ = n²π²ℏ²/(2mL²). Comparing with numerics validates our method.

73Numerical vs Exact Comparison

With 200 grid points, numerical eigenvalues match exact values to within 0.01% for the first few states. Error increases for higher states due to the finite grid.

108Finite Square Well

For more complex potentials without analytical solutions, the numerical method shines. The finite well has a finite number of bound states with E < 0.

125 lines without explanation

1import numpy as np
2import matplotlib.pyplot as plt
3from scipy.linalg import eigh_tridiagonal
4
5def solve_tise_finite_difference(V, L, N, m=1, hbar=1):
6    """
7    Solve the 1D time-independent Schrödinger equation:
8
9        -ℏ²/(2m) d²ψ/dx² + V(x)ψ = Eψ
10
11    using finite differences on a grid of N points.
12
13    Parameters:
14        V: function V(x) defining the potential
15        L: domain [0, L]
16        N: number of grid points
17        m: particle mass
18        hbar: reduced Planck constant
19
20    Returns:
21        x: grid points
22        energies: eigenvalues (sorted)
23        wavefunctions: eigenvectors (columns)
24    """
25    # Create spatial grid (excluding boundaries where ψ = 0)
26    dx = L / (N + 1)
27    x = np.linspace(dx, L - dx, N)
28
29    # Construct the Hamiltonian matrix using finite differences
30    # H = -ℏ²/(2m) d²/dx² + V(x)
31    #
32    # Second derivative: d²ψ/dx² ≈ (ψ_{i+1} - 2ψ_i + ψ_{i-1}) / dx²
33    #
34    # This gives a tridiagonal matrix with:
35    #   diagonal:     -2 / dx²  + V(x_i) * (2m/ℏ²)
36    #   off-diagonal: 1 / dx²
37
38    kinetic_factor = hbar**2 / (2 * m * dx**2)
39
40    # Tridiagonal matrix elements
41    diagonal = 2 * kinetic_factor + V(x)
42    off_diagonal = -kinetic_factor * np.ones(N - 1)
43
44    # Solve the eigenvalue problem H ψ = E ψ
45    # using specialized tridiagonal solver (much faster than general eig)
46    energies, wavefunctions = eigh_tridiagonal(diagonal, off_diagonal)
47
48    # Normalize wavefunctions: ∫|ψ|² dx = 1
49    for i in range(N):
50        norm = np.sqrt(np.sum(wavefunctions[:, i]**2) * dx)
51        wavefunctions[:, i] /= norm
52
53    return x, energies, wavefunctions
54
55def infinite_square_well_exact(n, L, m=1, hbar=1):
56    """
57    Exact solution for the infinite square well.
58
59    Eigenfunctions: ψ_n(x) = √(2/L) sin(nπx/L)
60    Eigenvalues:    E_n = n²π²ℏ²/(2mL²)
61    """
62    E_n = (n * np.pi * hbar)**2 / (2 * m * L**2)
63
64    def psi_n(x):
65        return np.sqrt(2/L) * np.sin(n * np.pi * x / L)
66
67    return E_n, psi_n
68
69# Example: Solve for an infinite square well
70L = 1.0  # Box width
71N = 200  # Grid points
72
73# V = 0 inside the box (boundaries enforce V = ∞ outside)
74V = lambda x: np.zeros_like(x)
75
76x, energies, wavefunctions = solve_tise_finite_difference(V, L, N)
77
78# Compare with exact solutions
79print("Comparing numerical vs exact eigenvalues:")
80print("-" * 45)
81for n in range(1, 6):
82    E_exact, _ = infinite_square_well_exact(n, L)
83    E_numerical = energies[n-1]
84    error = abs(E_numerical - E_exact) / E_exact * 100
85    print(f"n={n}: E_exact = {E_exact:8.4f}, E_num = {E_numerical:8.4f}, "
86          f"error = {error:.4f}%")
87
88# Visualize the first few eigenfunctions
89fig, axes = plt.subplots(2, 3, figsize=(14, 8))
90
91for n, ax in enumerate(axes.flat, 1):
92    # Numerical solution (add boundary points)
93    x_full = np.concatenate([[0], x, [L]])
94    psi_num = np.concatenate([[0], wavefunctions[:, n-1], [0]])
95
96    # Exact solution
97    E_exact, psi_exact_func = infinite_square_well_exact(n, L)
98    x_exact = np.linspace(0, L, 500)
99    psi_exact = psi_exact_func(x_exact)
100
101    # Match signs (eigenvectors can have arbitrary sign)
102    if np.sum(psi_num * np.interp(x_full, x_exact, psi_exact)) < 0:
103        psi_num = -psi_num
104
105    ax.plot(x_exact, psi_exact, 'b-', lw=2, label='Exact', alpha=0.7)
106    ax.plot(x_full, psi_num, 'r--', lw=2, label='Numerical')
107    ax.fill_between(x_exact, psi_exact, alpha=0.2)
108    ax.axhline(0, color='gray', lw=0.5)
109    ax.set_title(f'ψ_{n}(x), E_{n} = {n**2:.0f}E₁')
110    ax.set_xlabel('x')
111    ax.set_ylabel('ψ(x)')
112    ax.legend()
113    ax.grid(True, alpha=0.3)
114
115plt.suptitle('Infinite Square Well: Numerical vs Exact Eigenfunctions',
116             fontsize=14)
117plt.tight_layout()
118plt.show()
119
120# Now solve for a more interesting potential: finite square well
121def finite_square_well(x, V0=50, a=0.3):
122    """Finite square well centered in the box."""
123    L = x[-1] + x[1] - x[0]  # Recover L from grid
124    center = L / 2
125    return np.where(np.abs(x - center) < a/2, -V0, 0)
126
127x, energies, wavefunctions = solve_tise_finite_difference(
128    lambda x: finite_square_well(x, V0=50, a=0.4), L, N)
129
130print("\nFinite square well bound state energies:")
131bound_states = energies[energies < 0]
132for i, E in enumerate(bound_states, 1):
133    print(f"  n={i}: E = {E:.4f}")

Test Your Understanding

📝Test Your Understanding

Check your understanding of the time-independent Schrödinger equation

1.What does the time-independent Schrödinger equation Ĥψ = Eψ represent?

2.For an infinite square well of width L, how do the energy eigenvalues scale with the quantum number n?

3.How many nodes (points where ψ = 0, excluding boundaries) does the n-th eigenfunction of an infinite square well have?

4.What is the ground state energy of a particle in an infinite square well?

5.Why must the wave function be continuous at the boundaries of a finite potential well?

6.What happens to the energy levels of an infinite square well if you double the width L?

0/6 answered

Summary

In this section, we derived and explored the time-independent Schrödinger equation — the eigenvalue problem that determines the allowed energy levels and stationary states of quantum systems.

Key Equations

Name	Formula
Time-Independent Schrödinger Equation	Ĥψ = Eψ
Hamiltonian Operator	Ĥ = -ℏ²/(2m) d²/dx² + V(x)
Infinite Well Eigenfunctions	ψₙ(x) = √(2/L) sin(nπx/L)
Infinite Well Eigenvalues	Eₙ = n²π²ℏ²/(2mL²)
Full Stationary State	Ψ(x,t) = ψ(x)e^{-iEt/ℏ}

Key Takeaways

The TISE is an eigenvalue problem: we seek wave functions $\psi$ and energies $E$ satisfying $\hat{H}\psi = E\psi$
Energy quantization arises from boundary conditions — only certain wavelengths "fit" in the box
The ground state has non-zero energy (zero-point energy) due to the uncertainty principle
Higher energy eigenfunctions have more nodes — more wiggles mean higher kinetic energy
Eigenfunctions are orthogonal and form a complete basis for any quantum state
The TISE connects to eigenvalue problems in ML: PCA, spectral clustering, graph neural networks
Numerical methods convert the differential equation to a matrix eigenvalue problem solvable with linear algebra

The Core Message:

"The time-independent Schrödinger equation is an eigenvalue problem. Boundary conditions select which energies are allowed, leading to the quantization that underlies all of atomic and molecular physics."

Coming Next: In the next section, we'll explore the time-dependent Schrödinger equation in more detail, studying how quantum states evolve in time and introducing the concept of wave packet dynamics.