Wave Functions and the Schrodinger Equation

In the position representation, the state is described by a wave function $\psi(\mathbf{r}, t)$ Where $|\psi(\mathbf{r}, t)|^2$ is the probability density:

$P(\mathbf{r} \in [\mathbf{r}, \mathbf{r} + d\mathbf{r}]) = |\psi(\mathbf{r}, t)|^2\, d^3\mathbf{r}$

Normalisation: $\int_{-\infty}^{\infty} |\psi(\mathbf{r}, t)|^2\, d^3\mathbf{r} = 1$.

$i\hbar \frac{\partial \psi}{\partial t} = \hat{H}\psi = \left(-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}, t)\right)\psi$

For time-independent potentials $V(\mathbf{r})$Separate variables: $\psi(\mathbf{r}, t) = \phi(\mathbf{r}) e^{-iEt/\hbar}$:

$\hat{H}\phi = E\phi \quad \mathrm{i}.e., \quad -\frac{\hbar^2}{2m}\nabla^2\phi + V\phi = E\phi$

This is an eigenvalue problem: $E$ is the energy eigenvalue, $\phi$ is the energy eigenstate.

Properties of energy eigenstates:

1. Orthogonality. If $\hat{H}\phi_n = E_n\phi_n$ and $\hat{H}\phi_m = E_m\phi_m$ with $E_n \neq E_m$ then $\int \phi_n^*\phi_m\,dx = 0$ (since $\hat{H}$ is Hermitian).

2. Completeness. The energy eigenstates form a complete basis: any state can be expanded as $\psi(x,0) = \sum_n c_n\phi_n(x)$ where $c_n = \int \phi_n^*(x)\psi(x,0)\,dx$.

3. Stationary states. If $\psi(x,0) = \phi_n(x)$ Then $\psi(x,t) = \phi_n(x)e^{-iE_nt/\hbar}$. The probability density $|\psi|^2 = |\phi_n|^2$ is time-independent.

4. Reality of $\phi$. If $V(x)$ is real and there is no magnetic field, $\phi_n(x)$ can be chosen to be real. This is because if $\phi_n$ is a solution, so is $\phi_n^*$ And degenerate solutions can be combined into real linear combinations.

The probability current density is

$\mathbf{J} = \frac{\hbar}{2mi}(\psi^* \nabla\psi - \psi \nabla\psi^*)$

It satisfies the continuity equation: $\frac{\partial |\psi|^2}{\partial t} + \nabla \cdot \mathbf{J} = 0$ Expressing conservation of probability.

Derivation of the continuity equation. Start with the Schrodinger equation and its complex conjugate:

$i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\psi + V\psi$

$-i\hbar \frac{\partial \psi^*}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\psi^* + V\psi^*$

Multiply the first by $\psi^*$ and the second by $\psi$ Then subtract:

$i\hbar\!\left(\psi^*\frac{\partial \psi}{\partial t} + \psi\frac{\partial \psi^*}{\partial t}\right) = -\frac{\hbar^2}{2m}\!\left(\psi^*\nabla^2\psi - \psi\nabla^2\psi^*\right)$

The left-hand side is $i\hbar\,\partial|\psi|^2/\partial t$. The right-hand side is a divergence:

$\nabla \cdot (\psi^*\nabla\psi - \psi\nabla\psi^*) = \psi^*\nabla^2\psi - \psi\nabla^2\psi^*$

Therefore:

$i\hbar\frac{\partial |\psi|^2}{\partial t} = -\frac{\hbar^2}{2m}\nabla \cdot (\psi^*\nabla\psi - \psi\nabla\psi^*)$

Dividing by $i\hbar$:

$\frac{\partial |\psi|^2}{\partial t} + \nabla \cdot \left[\frac{\hbar}{2mi}(\psi^*\nabla\psi - \psi\nabla\psi^*)\right] = 0$

$\frac{\partial |\psi|^2}{\partial t} + \nabla \cdot \mathbf{J} = 0 \qquad \blacksquare$

Theorem 3.1. Time evolution governed by the Schrodinger equation with a Hermitian Hamiltonian Is unitary, and therefore preserves the norm of the state vector.

Proof. The time evolution operator $\hat{U}(t, t_0)$ is defined by:

$|\psi(t)\rangle = \hat{U}(t, t_0)|\psi(t_0)\rangle$

For a time-independent Hamiltonian:

$\hat{U}(t, t_0) = \exp\!\left(-\frac{i\hat{H}(t - t_0)}{\hbar}\right)$

To prove unitarity, we show $\hat{U}^\dagger \hat{U} = \hat{I}$:

$\hat{U}^\dagger = \exp\!\left(\frac{i\hat{H}^\dagger(t - t_0)}{\hbar}\right) = \exp\!\left(\frac{i\hat{H}(t - t_0)}{\hbar}\right)$

Since $\hat{H} = \hat{H}^\dagger$ (Hermitian). Therefore:

$\hat{U}^\dagger \hat{U} = \exp\!\left(\frac{i\hat{H}(t - t_0)}{\hbar}\right)\exp\!\left(-\frac{i\hat{H}(t - t_0)}{\hbar}\right) = \hat{I}$

Since commuting operators satisfy $e^A e^{-A} = I$.

Consequence. Norm preservation:

$\langle\psi(t)|\psi(t)\rangle = \langle\psi(t_0)|\hat{U}^\dagger\hat{U}|\psi(t_0)\rangle = \langle\psi(t_0)|\psi(t_0)\rangle$

Total probability is conserved under time evolution. $\blacksquare$

Composing evolutions. For successive time intervals, the evolution operator composes as:

$\hat{U}(t_2, t_0) = \hat{U}(t_2, t_1)\,\hat{U}(t_1, t_0)$

This composition law, combined with unitarity, is the group structure underlying quantum dynamics. For a time-dependent Hamiltonian, the evolution operator is given by Dyson”s time-ordered exponential:

$\hat{U}(t, t_0) = \mathcal{T}\exp\!\left(-\frac{i}{\hbar}\int_{t_0}^{t}\hat{H}(t')\,dt'\right)$

Where $\mathcal{T}$ denotes time ordering (later times appear to the left).

For a time-dependent Hamiltonian, the evolution operator satisfies $i\hbar\,\partial\hat{U}/\partial t = \hat{H}(t)\hat{U}$ With $\hat{U}(t_0, t_0) = \hat{I}$. Unitarity still holds: $d(\hat{U}^\dagger\hat{U})/dt = 0$ since $\hat{H}(t) = \hat{H}^\dagger(t)$.

A physically valid wave function must satisfy $\int |\psi|^2\,dx = 1$. This determines the normalisation Constant.

Example 3.1. Normalise the wave function $\psi(x) = Ae^{-\alpha|x|}$ for $-\infty \lt x \lt \infty$ Where $\alpha \gt 0$.

Example 3.2. Normalise $\psi(x) = Axe^{-\alpha x^2}$ for $-\infty \lt x \lt \infty$.

Problem. A particle of mass $m$ is confined to a one-dimensional box of width $L$ (infinite square well). Find the energy eigenstates and eigenvalues.

Problem. For the ground state of the infinite square well, find $\langle x \rangle$, $\langle x^2 \rangle$, $\langle p \rangle$, and $\langle p^2 \rangle$. Verify the uncertainty principle.

When the Hamiltonian has a time-dependent perturbation, $\hat{H}(t) = \hat{H}_0 + \hat{V}(t)$The Transition probability from initial state $|i\rangle$ to final state $|f\rangle$ (with $E_i \neq E_f$) is computed in the interaction picture.

First-order transition amplitude. If the system starts in $|i\rangle$ at $t = 0$The probability Amplitude for being in $|f\rangle$ at time $t$ is, to first order:

$c_f(t) = -\frac{i}{\hbar}\int_0^t \langle f | \hat{V}(t') | i \rangle\, e^{i\omega_{fi}t'}\,dt'$

Where $\omega_{fi} = (E_f - E_i)/\hbar$ is the Bohr frequency.

Constant perturbation. If $\hat{V}(t) = \hat{V}_0$ (constant) for $0 \lt t \lt T$:

$c_f(T) = -\frac{i}{\hbar}V_{fi}\int_0^T e^{i\omega_{fi}t'}\,dt' = -\frac{V_{fi}}{\hbar\omega_{fi}}\!\left(e^{i\omega_{fi}T} - 1\right)$

The transition probability is:

$P_{i \to f}(T) = \frac{|V_{fi}|^2}{\hbar^2}\,\frac{\sin^2(\omega_{fi}T/2)}{(\omega_{fi}/2)^2}$

This function is sharply peaked around $\omega_{fi} = 0$ (resonance), with width $\Delta\omega \sim 2\pi/T$.

Interpretation. As $T \to \infty$The function $\sin^2(\omega_{fi}T/2)/(\omega_{fi}/2)^2 \to 2\pi T\,\delta(\omega_{fi})$ So transitions occur only when energy is conserved ($E_f = E_i$). For finite $T$Energy conservation Is approximate to within $\Delta E \sim \hbar/T$A manifestation of the time-energy uncertainty Relation.

Fermi’s Golden Rule. For a transition to a continuum of final states with density of states $\rho(E_f)$The transition rate (probability per unit time) is:

$\Gamma_{i \to f} = \frac{2\pi}{\hbar}|\langle f | \hat{V} | i \rangle|^2\,\rho(E_f)$

This is one of the most important results in quantum mechanics, with applications to spontaneous Emission, scattering theory, and condensed matter physics.

Sudden and adiabatic approximations.

• Sudden approximation. If the Hamiltonian changes rapidly compared to the system’s natural timescale $\sim \hbar/\Delta E$The state does not have time to adjust: $|\psi_{\mathrm{after}\rangle = |\psi_{\mathrm{before}\rangle}}$. The probability of finding the system in the new $n$-th eigenstate is $P_n = |\langle n_{\mathrm{new}|\psi_{\mathrm{before}\rangle|^2}}$.

• Adiabatic theorem. If the Hamiltonian changes slowly enough (specifically, if $|\langle m|\partial\hat{H}/\partial t|n\rangle|/(\hbar\omega_{mn}^2) \ll 1$ for all $m \neq n$), the system remains in an instantaneous eigenstate without transitions. The adiabatic condition requires the rate of change to be much slower than the energy gap divided by $\hbar$.

Harmonic perturbation. For a sinusoidal perturbation $\hat{V}(t) = \hat{V}_1\,e^{-i\omega t} + \hat{V}_1^\dagger\,e^{i\omega t}$ The first-order transition rate from $|i\rangle$ to $|f\rangle$ is significant only when $\omega \approx \omega_{fi}$ (absorption) or $\omega \approx -\omega_{fi}$ (stimulated emission). The transition probability for Resonant absorption ($\omega \approx \omega_{fi}$) is:

$P_{i\to f}(t) = \frac{|\langle f|\hat{V}_1|i\rangle|^2}{\hbar^2}\,\frac{\sin^2((\omega - \omega_{fi})t/2)}{(\omega - \omega_{fi})^2/4}$

In the long-time limit, this reduces to Fermi’s Golden Rule with the replacement $V_{fi} \to \langle f|\hat{V}_1|i\rangle$.

flowchart TD
    A[3_Wave Functions And The Schrodinger Equation] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

The Schrodinger equation is the quantum version of Newton’s second law: it tells you how a quantum state evolves in time. The wave function is like a ghostly cloud that describes where a particle might be found, with the square of its height giving the probability. A particle in a box is like a vibrating string fixed at both ends: only certain wavelengths fit, so only certain energies are allowed. The zero-point energy is the quantum version of the jitterbug: you cannot pin a particle down completely because the uncertainty principle demands it keep moving. Probability current is like water flow: the continuity equation ensures that probability is never created or destroyed, only moved around. Unitarity guarantees that quantum mechanics is reversible: if you run time backward, you recover the initial state, just like rewinding a video.

• Postulates of Quantum Mechanics: The postulates provide the mathematical framework for wave functions and the Schrodinger equation.

• Operators and Observables: Operators represent physical observables and act on wave functions to extract measurable quantities.

• One-Dimensional Problems: The infinite square well and other 1D problems illustrate the application of the Schrodinger equation.

• Calculus

• Linear Algebra

• Vector Calculus

• Quantum Computing

Mistake 1: Forgetting to normalise the wave function A physically valid wave function must satisfy $\int |\psi|^2\,dx = 1$. Students often solve the Schrodinger equation and present the unnormalised solution. The normalisation constant is determined by this integral and affects all expectation values and probabilities. Always normalise before computing physical quantities.

Mistake 2: Applying incorrect boundary conditions For the infinite square well, $\psi = 0$ at the walls because the potential is infinite there. For the finite square well, $\psi$ and $d\psi/dx$ must be continuous at the boundary. For the delta function potential, $\psi$ is continuous but $d\psi/dx$ has a discontinuity. Using the wrong matching conditions gives incorrect energy eigenvalues.

Mistake 3: Assuming all wave functions are normalisable Plane waves $e^{ikx}$ are not normalisable over all space and cannot represent physical states by themselves. Physical states are wave packets constructed by superposing plane waves. The energy spectrum for free particles is continuous, and the eigenfunctions form a continuum rather than a discrete set.