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Chapter 2 of 14

Wavefunctions, Probability, and the Schrödinger Equation

A mysterious function ψ(x,t) suddenly claims to encode everything knowable about a particle—yet it is not directly observable. See how this abstract object becomes the central character of quantum theory and how a single differential equation dictates its fate.

15 min readen

From Classical Paths to Quantum Wavefunctions

Classical vs Quantum

Classically, a particle has a definite position x and momentum p at every instant. If you know both exactly, you can in principle predict its entire future path.

Experimental Trouble

Double-slit and related experiments show single electrons can build up interference patterns, behaving like waves and particles. The simple classical path picture breaks down.

Enter the Wavefunction

Quantum mechanics introduces the wavefunction ψ(x,t), a complex-valued function that does not give position directly but encodes probability amplitudes instead.

New Question

We shift from asking where the particle is to asking: given ψ(x,t), what is the probability of finding the particle near x if we measure now?

Wavefunction as Probability Amplitude and the Born Rule

Probability Amplitude

The wavefunction ψ(x,t) is a probability amplitude, not a probability itself. It can be complex-valued, with real and imaginary parts.

Born Rule

Born rule: the probability density of finding the particle at x at time t is P(x,t) = |ψ(x,t)|², where |ψ|² = ψψ and ψ is the complex conjugate.

From Density to Probability

The probability of finding the particle between x and x+dx at time t is |ψ(x,t)|² dx. The density |ψ|² has units of 1/length in 1D.

Why This Matters

This rule turns the abstract wavefunction into concrete predictions about measurement outcomes, but it also forces us to demand that total probability equals 1.

Thought Exercise: Interpreting |ψ|² Shapes

3. Thought Exercise: Interpreting |ψ|² Shapes

Imagine three different graphs of `|ψ(x,t₀)|²` for a particle at some fixed time `t₀`:

  1. Graph A: A single narrow peak around `x = 0`, very small everywhere else.
  2. Graph B: Two equal peaks: one near `x = -2 cm`, one near `x = +2 cm`, with almost zero in between.
  3. Graph C: A broad hump spread smoothly from `x = -5 cm` to `x = +5 cm`.

For each graph, answer mentally:

  • a) Where are position measurements most likely to find the particle?
  • b) Where are they unlikely to find it?
  • c) Which graph corresponds to the most localized particle?
  • d) Which graph corresponds to the least localized particle (most uncertain position)?

Now reason it out:

  • In all cases, higher `|ψ|²` at some x means measurements are more likely to give positions near that x.
  • A narrow, high peak means the particle is well localized.
  • A broad distribution means more uncertainty in position.

Try to connect this to the double-slit experiment:

  • Where `|ψ|²` has bright fringes, detections are likely.
  • Where `|ψ|²` is almost zero (dark fringes), detections are rare.

Normalization: Making Total Probability = 1

Total Probability

Because |ψ(x,t)|² is a probability density, the total probability of finding the particle somewhere must be 1: ∫_{-∞}^{+∞} |ψ(x,t)|² dx = 1.

How to Normalize

Given a non-normalized φ(x), compute N = ∫ |φ(x)|² dx and define ψ(x) = φ(x)/√N. Then ψ(x) is normalized and has total probability 1.

Finite Regions and 3D

If the particle is confined to 0 ≤ x ≤ L, normalize with ∫₀ᴸ |ψ(x)|² dx = 1. In 3D, the condition is ∫ |ψ(r⃗,t)|² d³r = 1.

Why It Matters

Normalization is essential: without it, |ψ|² could not consistently represent probabilities, since the total chance of finding the particle might not be 1.

Example: Normalizing a Simple Wavefunction

Setup

A particle is in a 1D box from x=0 to x=L with ψ(x) = A sin(πx/L) inside and 0 outside. Find A so that ψ is normalized.

Normalization Condition

Because ψ is nonzero only between 0 and L, we impose ∫₀ᴸ |ψ(x)|² dx = 1. Substituting ψ gives A² ∫₀ᴸ sin²(πx/L) dx = 1.

Integral and Result

Using ∫₀ᴸ sin²(πx/L) dx = L/2, we get A²(L/2) = 1, so A² = 2/L and A = √(2/L).

Normalized Wavefunction

The normalized state is ψ(x) = √(2/L) sin(πx/L) for 0 ≤ x ≤ L. This is the ground state of the infinite square well.

The Time-Dependent Schrödinger Equation

Evolution of ψ

To know how ψ(x,t) changes in time, we use the time-dependent Schrödinger equation, which plays a role similar to Newton's laws in classical mechanics.

Full Equation

In 1D with potential V(x,t): iħ ∂ψ/∂t = [-(ħ²/2m) ∂²/∂x² + V(x,t)] ψ(x,t). The bracketed operator is the Hamiltonian Ĥ.

Key Features

The equation is first order in time, second order in space, and linear. Linearity means any linear combination of solutions is also a solution.

Why It Matters

Given ψ(x,t₀) at some time, the Schrödinger equation fixes ψ at all other times. This makes ψ a complete dynamical description (within non-relativistic quantum theory).

Time-Independent Schrödinger Equation and Stationary States

Time-Independent Case

If V(x,t) = V(x) is time-independent, we can look for solutions of the form ψ(x,t) = φ(x)T(t). This leads to a spatial equation for φ(x).

Eigenvalue Equation

The spatial part satisfies [-(ħ²/2m) d²/dx² + V(x)] φ(x) = E φ(x), or Ĥφ = Eφ. This is the time-independent Schrödinger equation.

Stationary States

Solutions φₙ(x) with energy Eₙ give full states ψₙ(x,t) = φₙ(x) e^{-iEₙt/ħ}. Their probability density |ψₙ|² = |φₙ|² is time-independent.

Superpositions

General states are superpositions of stationary states. Their probability distributions can change in time due to interference between different energy components.

Boundary Conditions and Quantized Energies

Role of Boundary Conditions

Solving the Schrödinger equation requires boundary conditions from physical constraints, such as confinement in a box or binding in an atom.

Physical Requirements

We usually require ψ to be finite, continuous, and square-integrable (∫|ψ|² dx < ∞), so that probabilities are well-defined.

Infinite Square Well

For a box 0<x<L with infinite walls, ψ=0 outside. Continuity implies ψ(0)=0 and ψ(L)=0, giving sine-shaped solutions inside.

Quantized Energies

These conditions lead to discrete energies Eₙ = (n²π²ħ²)/(2mL²) and normalized states ψₙ(x) = √(2/L) sin(nπx/L). Boundary conditions cause quantization.

Quick Python Demo: Evolving a Superposition in a Box

9. Quick Python Demo: Evolving a Superposition in a Box

This short Python example (using standard scientific libraries) illustrates how a superposition of two stationary states in an infinite square well evolves in time.

We use units where `ħ = 1`, `m = 1`, and `L = 1` for simplicity.

```python

import numpy as np

import matplotlib.pyplot as plt

Parameters (ħ = m = L = 1 units)

L = 1.0

x = np.linspace(0, L, 500)

Stationary states for infinite square well (0 < x < L)

def psi_n(n, x):

return np.sqrt(2 / L) np.sin(n np.pi * x / L)

Energies E_n = n^2 * π^2 / (2mL^2); with m=L=1, this is:

def E_n(n):

return 0.5 (n np.pi) 2

Build a superposition of n=1 and n=2

c1, c2 = 1/np.sqrt(2), 1/np.sqrt(2) # equal weights

Time-dependent wavefunction ψ(x,t)

def psi_xt(x, t):

psi1 = c1 psi_n(1, x) np.exp(-1j E_n(1) t)

psi2 = c2 psi_n(2, x) np.exp(-1j E_n(2) t)

return psi1 + psi2

Choose a time and plot |ψ(x,t)|^2

for t in [0.0, 0.1, 0.2, 0.5]:

psi = psi_xt(x, t)

prob_density = np.abs(psi) 2

plt.plot(x, prob_density, label=f"t = {t:.2f}")

plt.xlabel("x")

plt.ylabel(r"|ψ(x,t)|²")

plt.title("Probability density in a superposition of n=1 and n=2")

plt.legend()

plt.tight_layout()

plt.show()

```

What you should notice if you run this:

  • Each individual stationary state has a time-independent `|ψₙ(x,t)|²`.
  • But their superposition has a probability density that oscillates in time.

This is a concrete visualization of how the time-dependent Schrödinger equation and stationary states work together.

Check Understanding: Wavefunctions and Schrödinger Equation

10. Check Understanding: Wavefunctions and Schrödinger Equation

Answer this question to test your grasp of the key ideas.

Which statement is correct about a normalized stationary state ψₙ(x,t) of a particle in a time-independent potential V(x)?

  1. Its probability density |ψₙ(x,t)|² oscillates in time, but the total probability remains 1.
  2. Its probability density |ψₙ(x,t)|² is independent of time, and the total probability is 1.
  3. Its wavefunction ψₙ(x,t) must be real and cannot be complex.
  4. Its energy can take any continuous value as long as ψₙ(x,t) is normalizable.
Show Answer

Answer: B) Its probability density |ψₙ(x,t)|² is independent of time, and the total probability is 1.

For a stationary state in a time-independent potential, ψₙ(x,t) = φₙ(x) e^{-iEₙt/ħ}. The phase factor has magnitude 1, so |ψₙ(x,t)|² = |φₙ(x)|² is time-independent. The state is normalized so the total probability is 1. The wavefunction can be complex, and boundary conditions usually restrict energies to discrete values, not a continuum, for bound states.

Review Key Terms

11. Review Key Terms

Use these flashcards to reinforce the main concepts from this module.

Wavefunction ψ(x,t)
A complex-valued function that encodes the quantum state of a system. It is a probability amplitude; |ψ(x,t)|² gives the probability density of finding a particle at position x at time t.
Born rule
The rule that connects the wavefunction to measurement outcomes: the probability density of finding a particle at x at time t is |ψ(x,t)|².
Normalization
The requirement that the total probability of finding the particle somewhere equals 1, expressed as ∫ |ψ(x,t)|² dx = 1 (or the appropriate multidimensional integral).
Time-dependent Schrödinger equation
The fundamental equation governing the time evolution of ψ: iħ ∂ψ/∂t = Ĥψ, where Ĥ is the Hamiltonian operator.
Time-independent Schrödinger equation
The eigenvalue equation Ĥφ = Eφ obtained for time-independent potentials V(x). Its solutions φ(x) give stationary states with definite energy E.
Stationary state
A state with definite energy E for which the probability density |ψ(x,t)|² does not change in time. It has the form ψ(x,t) = φ(x) e^{-iEt/ħ}.
Boundary conditions
Physical requirements imposed on ψ (such as being finite, continuous, zero at infinite walls) that restrict allowed solutions and lead to quantized energies in bound systems.
Hamiltonian operator Ĥ
The operator representing the total energy of the system. In 1D, Ĥ = - (ħ²/2m) d²/dx² + V(x). Its eigenvalues are the allowed energies.

Key Terms

Born rule
The postulate that the probability density of obtaining a measurement result is given by the modulus squared of the corresponding probability amplitude, such as |ψ(x,t)|² for position.
Hamiltonian
The operator representing the total energy (kinetic plus potential) of a quantum system. Its eigenvalues correspond to possible measured energies.
wavefunction
A complex-valued function ψ(x,t) (or ψ(r⃗,t)) that fully describes the quantum state of a system in non-relativistic quantum mechanics. Its modulus squared gives probability densities.
normalization
The condition that the total probability of all possible outcomes is 1. For a wavefunction, this is expressed as an integral of |ψ|² over all space equaling 1.
stationary state
An energy eigenstate whose probability density is independent of time, even though the wavefunction itself may acquire a time-dependent phase factor.
boundary conditions
Constraints on the wavefunction at the edges of the region or at infinity, imposed by physical requirements such as confinement or normalizability.
energy quantization
The phenomenon that, for many bound quantum systems, only discrete energy levels are allowed, often arising from boundary conditions and normalizability.
Schrödinger equation
The fundamental differential equation of non-relativistic quantum mechanics. The time-dependent form iħ ∂ψ/∂t = Ĥψ governs how the wavefunction evolves in time.
probability amplitude
A complex quantity whose modulus squared gives a probability or probability density. The wavefunction is a probability amplitude for position and other observables.
time-dependent Schrödinger equation
The version of the Schrödinger equation that explicitly involves time derivatives of the wavefunction and describes its full time evolution.
time-independent Schrödinger equation
The eigenvalue equation Ĥφ = Eφ obtained when the potential is time-independent. It yields stationary states with definite energies.

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