The Schrödinger Equation and the Wavefunction

The exciting journey into the microscopic world¶

• In the next few sections, we will introduce Schrödinger’s Equation (SE), one of the fundamental laws of physics. A “fundamental law” means that SE cannot be derived from more basic principles; it can only be inferred or hypothesized based on experimental evidence. Its validity is supported by countless successful quantitative predictions and explanations of experimental observations.

• It’s important to emphasize that there has never been an instance where quantum mechanics has failed when applied correctly. The physical world is inherently quantum, especially at small scales. Quantum mechanics works flawlessly at all scales and in all situations where it has been properly applied.

What do we require from the new quantum theory?¶

• Recall that while classical mechanics is valid at large scales, it completely fails to describe motion at the atomic and molecular levels. A new, accurate equation of motion is required to explain phenomena such as:

  • The quantized nature of energy, as observed in experiments involving blackbody radiation and atomic and molecular spectra.

  • Wave-particle duality, demonstrated through electron diffraction, Compton scattering, and double-slit experiments.

Quantum wave equation¶

• In 1925/1926 Erwin Schrödinger, an expert on the physics of waves, derived a new equation of motion that predicts quantum phenomena such as energy quantization and wave-particle duality from first principles.

• We can trace Schrödinger’s approach by starting with the classical wave equation:

  $$\frac{\partial^2 \Psi(x,t)}{\partial x^2}=\frac{1}{v^2}\frac{\partial^2 \Psi(x,t)}{\partial t^2}$$

  (1)

• The classical wave equation we have seen can produce traveling or standing waves depending on boundary conditions. Let us pick a general periodic traveling wave, for instance:

• We are going to plug into a wave function the two key quantum relations discovered empirically and then see what kind of wave equation can produce it:

  • Plug wave-particle duality via De Broglie relation:

  $$p=h/\lambda=\hbar k\,\,\,\, where\,\,\,\, k=\frac{2\pi}{\lambda}$$

  (3)

  • Plug energy quantization via Planck equation:

  $$E=h\nu=\hbar\omega\,\,\,\, where\,\,\,\, \omega=2\pi\nu$$

  (4)

• Quantum wave function

• What equation can generate such quantum wave functions? To find out we need to take derivatives with respect to time and space.

From Quantum Wave Function to Quantum Wave Equation¶

• Time Part: When we take the time derivative of the wave function, we find that the total energy appears as a multiplicative factor. This is significant because, in quantum mechanics, total energy is conserved. The relationship between energy and the time dependence of the wave function is given by:

• Spatial Part: To recover the total energy from the spatial part of the wave function, we take two spatial derivatives.

• This equation, in which the time variable is absent, is the time-independent Schrödinger equation. We will come back to it shortly.

• Joining time and spatial parts by expressing the kinetic energy as a difference between total and potential energies $K = E-V$ and eliminating $E$ we get:

Quantum vs Classical Wave equation¶

• The Schrödinger equation describes the evolution of the wave function $\Psi(x,t)$ for a quantum system such as an electron or atom in a potential $V(x)$.

• Unlike the classical wave equation, there is only a single time derivative. The presence of $i$ generates oscillatory solutions in the complex plane, which is why we call it a quantum wave equation.

• What is the meaning of $\Psi(x,t)$? It is generally complex and so cannot stand for any real measurable quantity. We will see how to extract information from $\Psi$ later.

• Note that we are focusing on the 1D case for simplicity. Generalizing to 3D involves adding similar terms that depend on the $y$ and $z$ coordinates of every quantum object.

Separation of Variables and the Time-Independent Schrödinger Equation¶

• By assuming the wave function can be separated into a product of a spatial part and a time part, $\Psi(x,t) = \psi(x) T(t)$, we can solve for each part independently:

• The time part $T(t)=e^{-iEt/\hbar}$ yields an oscillatory solution related to the total energy.

• The spatial part $\psi(x)$ satisfies the time-independent Schrödinger equation, which we solve to obtain stationary states.

• This method is crucial for solving quantum systems, particularly in cases where the potential $V(x)$ does not depend on time.

• Plugging in $\psi(x) T(t)$ and cancelling the time part, we are back to the time-independent equation we obtained earlier.

From the equation to its meaning¶

We now have the equation and a strategy for solving it. What we do not yet have is the meaning of the object the equation keeps handing us. This half of the lecture answers that question and turns $\psi$ into numbers an experiment can check.

What is the meaning of a wave-function $\psi$ ?¶

• In the classical wave equation, the wave function has a clear mechanical interpretation: it represents the degree of disturbance in the wave. For example, it can describe the elevation of a guitar string from its resting position.

• In contrast, the quantum wave function is less intuitive. The wave function itself does not have direct physical meaning, as it is generally a complex function. To connect it to measurable quantities, we need to extract real values from it that correspond to physical observables.

• The key insight is that the absolute square of the wave function gives the probability distribution:

• $p(x)$ is a probability distribution function. It describes the likelihood of finding a quantum object at a position $x$.

• $p(x)\,dx$ gives the probability of finding the particle in a tiny part of space inside the interval $[x, x+dx]$.

• In three-dimensional space, the analogous expression is:

  $$p(x, y, z) = \psi(x, y, z)^{*} \cdot \psi(x, y, z)$$

  (13)

Probability Refresher¶

• Before introducing quantum mechanics and wavefunctions, let’s recall some core ideas from probability.

Random Variables¶

• A random variable assigns numbers to the outcomes of an experiment. For example, how many squirrels you see each day is a random variable.

• Discrete examples: dice rolls, coin flips.

• Continuous examples: particle position, measurement noise.

Probability Distributions¶

• A continuous random variable is fully described by a distribution over all possible values. We call this object a probability distribution $p(x)$, which must integrate (or sum, in the discrete case) to one, showing that we cover all possibilities and that each possibility is assigned a fraction of 1.

Further Exploration¶

• Video overview

• Interactive probability explorer

Source

import matplotlib.pyplot as plt
import numpy as np

# Discrete probabilities (fair dice)
outcomes = np.arange(1, 7)
probs = np.ones_like(outcomes) / 6

plt.bar(outcomes, probs)
plt.xlabel("Dice outcome")
plt.ylabel("Probability")
plt.title("PMF of a Fair Die")
plt.show()

Source

# -----------------------------
# Correct sampling for H-atom 1s
# -----------------------------
N = 6000
a0 = 1.0

# For 1s: if x = 2r/a0, then x ~ Gamma(k=3, theta=1)
x = np.random.gamma(shape=3.0, scale=1.0, size=N)
r = 0.5 * a0 * x

# Isotropic angles
u = np.random.rand(N)
theta = np.arccos(1 - 2*u)
phi = 2*np.pi*np.random.rand(N)

# Convert to Cartesian and take a 2D projection
x3 = r * np.sin(theta) * np.cos(phi)
y3 = r * np.sin(theta) * np.sin(phi)

# Histogram for the shell probability density P(r) (per unit r)
bins = np.linspace(0, 6*a0, 80)
hist, edges = np.histogram(r, bins=bins, density=True)
centers = 0.5*(edges[1:] + edges[:-1])

# Analytical radial distribution P(r) = 4 r^2 |psi|^2 = 4 r^2 / (a0^3*pi) * exp(-2r/a0) * pi
# Simplifies to: P(r) = (4/a0^3) r^2 exp(-2r/a0), which integrates to 1
rr = np.linspace(0, bins[-1], 400)
P_r = (4.0/(a0**3)) * rr**2 * np.exp(-2*rr/a0)

# Side-by-side subplots: dot cloud and radial distribution

fig, axs = plt.subplots(1, 2, figsize=(12, 5))

# Left: simulated dot cloud
axs[0].scatter(x3, y3, s=1, alpha=0.5)
axs[0].set_title("Hydrogen 1s: position observations (2D projection)")
axs[0].set_xlabel("x [$a_0$]")
axs[0].set_ylabel("y [$a_0$]")
axs[0].set_aspect('equal')

# Right: radial distribution histogram + analytical curve
axs[1].plot(centers, hist, lw=2, label="Monte Carlo (histogram)")
axs[1].plot(rr, P_r, lw=2, linestyle="--", label="Analytical 1s")
axs[1].set_title("Radial probability distribution")
axs[1].set_xlabel("r [$a_0$]")
axs[1].set_ylabel("$P(r)$")
axs[1].legend()

plt.tight_layout()
plt.show()

Normalization of wavefunction¶

• For the wave function to represent a proper probability distribution, it must be normalizable. If it is not normalizable, the wave function is only proportional to a probability distribution and not equal to it.

• Normalization of $\psi^2$ ensures that there is absolute certainty that the quantum object exists somewhere in space. In an experiment, when searching for a quantum particle across the entire space, normalization guarantees that you will find it somewhere.

• Normalization in 1D:

  $$\int^{+\infty}_{-\infty} |\psi(x)|^2 dx = \int^{+\infty}_{-\infty} p(x) dx = 1$$

  (33)

• To normalize a wave function $\psi'$, multiply it by a constant: $\psi = N\psi'$. The constant $N$ is determined by plugging this expression into the normalization condition. In other words, normalization determines the multiplicative factor in front of the wave function.

• Normalization in 3D:

  $$\int^{+\infty}_{-\infty} \int^{+\infty}_{-\infty} \int^{+\infty}_{-\infty} |\psi(x, y, z)|^2 dx \, dy \, dz = 1$$

  (34)

Source

import numpy as np
import matplotlib.pyplot as plt

# Define x values
x = np.linspace(0, 1, 500)

# Unnormalized psi
psi_unnorm = x
psi2_unnorm = psi_unnorm**2

# Normalized psi (C = sqrt(3))
psi_norm = np.sqrt(3) * x
psi2_norm = psi_norm**2

# Compute integrals for annotation
area_unnorm = np.trapezoid(psi2_unnorm, x)
area_norm = np.trapezoid(psi2_norm, x)

# Create the figure and axes
fig, axes = plt.subplots(1, 2, figsize=(12, 5))

# --- Plot 1: Unnormalized psi² ---
axes[0].fill_between(x, psi2_unnorm, alpha=0.4, color="blue")
axes[0].set_title("Unnormalized $|\psi(x)|^2$")
axes[0].set_xlabel("x")
axes[0].set_ylabel(r"$|\psi(x)|^2$")
axes[0].grid(True)
axes[0].text(0.5, 0.2, rf"Area = {area_unnorm:.2f}", 
             transform=axes[0].transAxes, fontsize=12, color="blue")

# --- Plot 2: Normalized psi² ---
axes[1].fill_between(x, psi2_norm, alpha=0.4, color="green")
axes[1].set_title("Normalized $|\psi(x)|^2$")
axes[1].set_xlabel("x")
axes[1].set_ylabel(r"$|\psi(x)|^2$")
axes[1].grid(True)
axes[1].text(0.5, 0.2, rf"Area = {area_norm:.2f}", 
             transform=axes[1].transAxes, fontsize=12, color="green")

plt.tight_layout()
plt.show()

<>:24: SyntaxWarning: invalid escape sequence '\p'
<>:33: SyntaxWarning: invalid escape sequence '\p'
<>:24: SyntaxWarning: invalid escape sequence '\p'
<>:33: SyntaxWarning: invalid escape sequence '\p'
/tmp/ipykernel_2783/3267431332.py:24: SyntaxWarning: invalid escape sequence '\p'
  axes[0].set_title("Unnormalized $|\psi(x)|^2$")
/tmp/ipykernel_2783/3267431332.py:33: SyntaxWarning: invalid escape sequence '\p'
  axes[1].set_title("Normalized $|\psi(x)|^2$")

What can we do with probability distribution functions (PDF)?¶

• By definition, the probability distribution function $p(x)$ lets us quantify the probability that a quantum “particle” is located in an infinitesimal slice $[x, x+dx]$ around the point $x$. This then enables us to find the probability in any finite region $[a,b]$ simply by integrating:

  $$p(a<x<b)=\int_a^b |\psi(x)|^2dx$$

  (41)

• In higher dimensions, e.g. 3D, we can locate the particle within a volume element $dx\,dy\,dz$, or any finite volume, via a similar integration:

  $$p(a_x<x<b_x,a_y<y<b_y, a_z<z<b_z )=\int^{b_x}_{a_x} \int^{b_y}_{a_y} \int^{b_z}_{a_z} |\psi(x,y,z)|^2dx dy dz$$

  (42)

The Mathematical Language of Quantum Mechanics: Operators¶

• To streamline our discussion and draw analogies with classical intuition, we need to introduce some essential notation and terminology.

• In quantum mechanics, we define operators as mathematical entities that transform one function into another. An operator can perform various actions on a function, such as differentiation, integration, addition, multiplication, and more. In this way, operators serve as tools to manipulate wave functions and extract physical information.

Linear operators¶

• Linear means that an operator acting on a sum of functions does not change the power of any of the functions.

• The Schrödinger equation is a linear differential equation. Hence it can be written as a linear operator acting on a wave function.

Schrödinger equation in operator notation¶

• By expressing the equation in operator notation, we can start to recognize various terms and see that the Schrödinger equation, like any proper equation of motion, embodies the principle of total energy conservation:

• Let us take the Schrödinger equation and denote the differentials as operators acting on the wave function. For example, let us look at the two sides of the time-dependent Schrödinger equation: $\left[ -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V(x) \right] \Psi$ and $i\hbar \frac{\partial }{\partial t} \Psi$

• The operator of the position part is called the Hamiltonian operator, and it has a deep connection to energy conservation and the Hamiltonian function of classical mechanics.

• Note how the potential energy term appears in the same form across different systems: it is always a function of spatial coordinates. Some examples of potentials:

  • $V(x) = 0$ represents a free particle,

  • $V(x) = kx^2$ describes a particle trapped in a harmonic potential,

  • $V(x) = \cos(x)$ corresponds to a particle in a periodic potential.

The correspondence principle of Quantum Mechanics¶

• Thanks to the universality of the energy conservation law, for every observable in classical mechanics there is a corresponding operator in quantum mechanics! Let us list them here:

What about quantities which correspond to operators?¶

• Recall that the mean value of $x$ is computed by weighting its values by their probabilities. For example, think of the average mass of a box of candies: we multiply the probability (or fraction) of each candy type by its mass and sum.

• Likewise, you can compute the average of any function of $x$, say $x^2$ or $\sin(x)$.

• For quantities like momentum or total energy, which are no longer simple functions as in classical mechanics but operators, $\hat{p}$ and $\hat{H}$, we simply use the operators in the definition of the moments:

Source

import numpy as np
import matplotlib.pyplot as plt
from math import erf, sqrt

# ----- Parameters -----
sigma = 1.0  # width of the Gaussian (harmonic oscillator ground state)
x0, x1 = -1.0, 1.0  # interval to integrate over

# ----- Define |psi(x)|^2 for a normalized Gaussian -----
def prob_density(x, sigma):
    # |psi(x)|^2 = (1/(sqrt(2*pi)*sigma)) * exp(-x^2/(2*sigma^2))
    return (1.0/(np.sqrt(2*np.pi)*sigma)) * np.exp(-x**2/(2*sigma**2))

# Probability via the error function (analytic CDF of a normal distribution)
def interval_probability(a, b, sigma):
    return 0.5*(erf(b/(sqrt(2)*sigma)) - erf(a/(sqrt(2)*sigma)))

P = interval_probability(x0, x1, sigma)

# ----- Plot -----
xs = np.linspace(-4*sigma, 4*sigma, 800)
ys = prob_density(xs, sigma)

fig = plt.figure(figsize=(10, 5))

# Title with the integral expression
plt.title(r"$P_{x_0<x<x_1}=\int_{x_0}^{x_1}|\psi(x)|^2\,dx$" + 
          f"\nSelected interval: [{x0:.2f}, {x1:.2f}]",
          loc="left")

# Curve
plt.plot(xs, ys, linewidth=3)

# Shade between x0 and x1 under the curve
mask = (xs >= x0) & (xs <= x1)
plt.fill_between(xs[mask], ys[mask], 0, alpha=0.25)

# Mark the boundaries with vertical dashed lines
plt.axvline(x0, linestyle="--", linewidth=2)
plt.axvline(x1, linestyle="--", linewidth=2)

# Tick marks and labels
plt.xlabel("x")
plt.ylabel(r"$|\psi(x)|^2$")

# Annotate the probability
plt.text(0.02, 0.90, rf"$P_{{{x0:.2f}<x<{x1:.2f}}} = {P:.3f}$",
         transform=plt.gca().transAxes, fontsize=16)

plt.xlim(xs.min(), xs.max())
plt.ylim(bottom=0)
plt.grid(True, alpha=0.3)
plt.tight_layout()

plt.show()

Linearity and the Principle of Superposition¶

• As we recall from solving the classical wave equation, whenever boundary conditions are imposed on the spatial domain of our PDE we can end up with an infinite number of solutions $u_n(x)$ discretized by integers $n=1,2,...$ for each spatial coordinate.

• The general solution was written as a linear combination of normal modes. Likewise, boundary conditions produce an infinite number of solutions to the quantum wave equation discretized by integers $n$: $\hat{H} \psi_n(x)=E_n \psi_n(x)$

• The general solution is again written as a linear combination of quantum wave functions, thanks to the linearity of the Schrödinger equation:

Eigenvalues and Eigenfunctions¶

• In both the classical wave equation and the time-independent Schrödinger equation, we can use operator notation to frame the problem as one of finding special functions and their corresponding multiplicative factors that satisfy specific operators.

• In general, the action of an operator can change a function, but in quantum mechanics, we are particularly interested in operators that preserve the form of the function, producing only a constant multiplicative factor.

• The time-independent Schrödinger equation can be viewed as an eigenfunction-eigenvalue problem.

Examples of using probabilistic calculations in QM¶

Example 1: Probability in a region (1D)¶

Take the normalized wavefunction on $[0,1]$: $\psi(x)=\sqrt{3}\,x$. Then $p(x)=|\psi(x)|^2=3x^2$.

• Probability that the particle lies in $[a,b]\subset[0,1]$:

• Concrete numbers, say $[0.3,0.6]$:

Example 2: 3D slice probability (separable state)¶

Let $\psi(x,y,z)=\sqrt{27}\,xyz$ on the unit cube $[0,1]^3$ (this is normalized since $|\psi|^2=27x^2y^2z^2$ and $\int_0^1 x^2dx=\tfrac13$, so $27(\tfrac13)^3=1$).

• Probability the particle is inside the rectangular box $[a_x,b_x]\times[a_y,b_y]\times[a_z,b_z]$:

Example 3: Averages and variance from a PDF¶

For $\psi(x)=\sqrt{3}\,x$ on $[0,1]$ (so $p(x)=3x^2$):

• Mean position:

• Mean square position:

• Variance and standard deviation:

Example 4: Operator expectations (momentum/energy in a box)¶

Operator rules:

• $\hat{p}=-i\hbar\,\dfrac{d}{dx}$,

• $\hat{H}=-\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2}+V(x)$.

Use the infinite square well on $[0,1]$ with $V(x)=0$ inside and $\psi_n(x)=\sqrt{2}\sin(n\pi x)$, which is normalized.

• Momentum expectation:

(the integrand is odd over a full sine half-wave, or integrate by parts with vanishing boundary terms).

• Momentum squared:

• Energy (since $\hat{H}=\hat{p}^2/2m$ inside the well):

Problems¶

Problem 1¶

Confirm that the following wavefunctions are eigenfunctions of linear momentum and kinetic energy (or neither or both):

• $C sin(ax)$

• $N e^{-ix/\hbar}$

Problem 2: Taking the Square of an Operator¶

Consider the operator $\hat{A} = x \frac{d}{dx}$. Find $\hat{A}^2$, i.e., $\hat{A}(\hat{A}f(x))$, and apply it to an arbitrary function $f(x)$.

Problem 3: Verifying Eigenfunction and Eigenvalue¶

Consider the operator $\hat{B} = -i\hbar \frac{d}{dx}$ (momentum operator). Verify that $f(x) = e^{ikx}$ is an eigenfunction of $\hat{B}$, and find the corresponding eigenvalue.

Problem 4: Testing for an Eigenfunction and Eigenvalue¶

Given the operator $\hat{C} = \frac{d^2}{dx^2}$ (second derivative operator), check if $f(x) = e^{-\alpha x^2}$ is an eigenfunction of $\hat{C}$, and find the eigenvalue if it is.

Problem 5: Linearity and Eigenfunction Testing¶

Consider the operator $\hat{D} = x^2 \frac{d}{dx}$. Show whether this operator is linear and check if $f(x) = x^n$ is an eigenfunction of $\hat{D}$.

Problem 6: Normalize and locate¶

Normalize $\psi(x) = \cos(\pi x / 2)$ on the interval $[-1, 1]$, then compute the probability of finding the particle in $[0, \tfrac{1}{2}]$.

Problem 7: Symmetry does the integrals¶

For the particle-in-a-box state $\psi_2(x) = \sqrt{2}\sin(2\pi x)$ on $[0, 1]$, evaluate $\langle x \rangle$ and $\langle p \rangle$, and explain both results using symmetry alone.