Quantum mechanics starts from a core postulate: the state of a quantum system is completely described by a wave function $\Psi(\mathbf{r}, t)$ . This wave function evolves in time according to the time-dependent Schrödinger equation:

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

The Hamiltonian operator $\hat{H}$ represents the total energy of the system. It's the sum of kinetic and potential energy operators:

$\hat{H} = \hat{T} + \hat{V}$

The kinetic energy operator is $\hat{T} = -\frac{\hbar^2}{2m}\nabla^2$ , where $\nabla^2$ is the Laplacian and $m$ is the particle mass. This operator encodes how the curvature of the wave function relates to the particle's kinetic energy.
The potential energy operator $\hat{V}$ depends on the system you're studying. For a harmonic oscillator it's quadratic in position; for a hydrogen atom it's the Coulomb potential. Whatever forces act on the particle, they enter through $\hat{V}$ .

Putting it all together for a single particle in one dimension, the full equation reads:

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

Notice that this equation is first-order in time but second-order in space. The factor of $i$ on the left side is what makes quantum mechanics fundamentally different from classical wave equations: it produces oscillatory phase evolution rather than simple exponential growth or decay.

Time-Independent Schrödinger Equation

When the potential $V$ doesn't depend on time, you can simplify things enormously through separation of variables. Assume the wave function factors into a spatial part and a temporal part:

$\Psi(\mathbf{r}, t) = \psi(\mathbf{r})\,\phi(t)$

Substituting this into the time-dependent equation and dividing both sides by $\psi(\mathbf{r})\,\phi(t)$ , the left side depends only on $t$ and the right side depends only on $\mathbf{r}$ . The only way both sides can be equal for all values of $\mathbf{r}$ and $t$ is if each side equals the same constant. That constant turns out to be the energy $E$ . This gives two equations:

The time part solves to $\phi(t) = e^{-iEt/\hbar}$ , a simple oscillating phase factor.
The spatial part satisfies the time-independent Schrödinger equation:

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

This is an eigenvalue equation: the Hamiltonian acts on $\psi$ and returns the same function multiplied by the energy $E$ . The allowed values of $E$ are the energy eigenvalues, and the corresponding functions $\psi(\mathbf{r})$ are the eigenfunctions. These represent stationary states, meaning the probability density $|\Psi(\mathbf{r},t)|^2 = |\psi(\mathbf{r})|^2$ doesn't change with time, even though the full wave function still carries that oscillating phase factor $e^{-iEt/\hbar}$ .

Solving Schrödinger's Equation for Simple Systems

Particle in a Box

The particle in a box is the simplest solvable quantum system and the best place to build intuition for quantization. A particle of mass $m$ is confined to a one-dimensional region of length $L$ , with infinitely high potential walls at $x = 0$ and $x = L$ . Inside the box, $V = 0$ .

Solving step by step:

Inside the box, the time-independent Schrödinger equation reduces to $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi$ .
This is a standard second-order ODE with general solution $\psi(x) = A\sin(kx) + B\cos(kx)$ , where $k = \sqrt{2mE}/\hbar$ .
Apply the boundary condition $\psi(0) = 0$ : this forces $B = 0$ , leaving $\psi(x) = A\sin(kx)$ .
Apply $\psi(L) = 0$ : this requires $\sin(kL) = 0$ , so $kL = n\pi$ where $n = 1, 2, 3, \ldots$
Solve for the allowed energies:

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

Normalize the wave function (require $\int_0^L |\psi|^2 dx = 1$ ) to get:

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

Key features to notice: the energy scales as $n^2$ , so the spacing between levels increases with $n$ . Each wave function $\psi_n$ has $n-1$ nodes (zero crossings inside the box). The ground state ( $n=1$ ) has zero nodes, the first excited state ( $n=2$ ) has one node, and so on. These are standing waves, just like vibrations on a guitar string.

Harmonic Oscillator

The quantum harmonic oscillator models any system near a potential energy minimum, making it one of the most widely applied models in chemistry. The potential is $V(x) = \frac{1}{2}kx^2$ , where $k$ is the force constant.

The time-independent Schrödinger equation for this system is:

$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2}kx^2\psi = E\psi$

This can be solved using either the power series (Frobenius) method or the algebraic ladder operator approach. Both yield the same results:

Energy eigenvalues:

$E_n = \left(n + \frac{1}{2}\right)\hbar\omega, \quad n = 0, 1, 2, \ldots$

where $\omega = \sqrt{k/m}$ is the classical angular frequency. The levels are evenly spaced by $\hbar\omega$ , in contrast to the particle in a box.

Zero-point energy: Even the ground state ( $n = 0$ ) has energy $\frac{1}{2}\hbar\omega$ . The particle can never be perfectly at rest at the bottom of the well. This is a direct consequence of the uncertainty principle.
Wave functions: Each $\psi_n(x)$ is a Hermite polynomial $H_n(\xi)$ multiplied by a Gaussian envelope $e^{-\xi^2/2}$ , where $\xi = x\sqrt{m\omega/\hbar}$ is a dimensionless coordinate. The Gaussian ensures the wave function decays to zero at large $|x|$ , but notably it doesn't reach zero at the classical turning points. There's a nonzero probability of finding the particle in the classically forbidden region.

The harmonic oscillator is the go-to model for molecular vibrations (IR spectroscopy) and phonons in solids.

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Interpretation of the Wave Function

Probability Density

The wave function $\Psi(\mathbf{r}, t)$ is complex-valued and not directly observable. What is observable is its squared modulus, $|\Psi(\mathbf{r}, t)|^2$ , which gives the probability density for finding the particle at position $\mathbf{r}$ at time $t$ .

The probability of finding the particle in a small volume element $d\mathbf{r}$ is:

$P(\mathbf{r}, t)\,d\mathbf{r} = |\Psi(\mathbf{r}, t)|^2\,d\mathbf{r}$

For this interpretation to be consistent, the wave function must be normalized:

$\int_{-\infty}^{\infty} |\Psi(\mathbf{r}, t)|^2\,d\mathbf{r} = 1$

This just says the particle must be somewhere with 100% certainty. A key result: the time-dependent Schrödinger equation preserves normalization. If you start with a normalized wave function, it stays normalized as it evolves.

Phase and Interference

The complex phase of the wave function, $\arg(\Psi)$ , has no direct physical meaning on its own. You can multiply any wave function by a global phase factor $e^{i\alpha}$ and all observable predictions remain unchanged.

However, relative phases between different components of a superposition are physically crucial. When two wave functions overlap, they interfere:

Constructive interference occurs where the components are in phase, increasing $|\Psi|^2$ .
Destructive interference occurs where they're out of phase, decreasing $|\Psi|^2$ .

This is exactly what produces the fringe pattern in the double-slit experiment. The interference pattern arises because the wave function passes through both slits simultaneously, and the two contributions have position-dependent relative phases when they recombine at the detector.

Properties of Stationary States

Energy Eigenvalues and Eigenfunctions

Stationary states are solutions to $\hat{H}\psi = E\psi$ . Each one has a definite energy $E$ , and a measurement of energy on that state will always return that value with certainty.

The eigenfunctions describe the spatial probability distribution of the particle. The ground state (lowest $E$ ) typically has no nodes and the broadest spatial distribution. Excited states have progressively more nodes and higher energies.

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Orthogonality and Completeness

Two properties of the set of eigenfunctions make them extremely useful mathematically:

Orthogonality: Different eigenfunctions are orthogonal, meaning:

$\int \psi_m^*(\mathbf{r})\,\psi_n(\mathbf{r})\,d\mathbf{r} = 0 \quad \text{for } m \neq n$

This guarantees that the stationary states are linearly independent. Physically, it means a system in state $\psi_n$ has zero probability of being measured in a different stationary state $\psi_m$ .

Completeness: The eigenfunctions form a complete basis for the Hilbert space. Any well-behaved wave function can be expanded as a linear combination of stationary states:

$\Psi(\mathbf{r}, t) = \sum_n c_n\,\psi_n(\mathbf{r})\,e^{-iE_n t/\hbar}$

The coefficients $c_n$ are found by projection: $c_n = \int \psi_n^*(\mathbf{r})\,\Psi(\mathbf{r}, 0)\,d\mathbf{r}$ .

Expectation Values

For a system in a stationary state, the expectation value of any time-independent observable $\hat{A}$ is itself time-independent:

$\langle \hat{A} \rangle = \int \psi^*(\mathbf{r})\,\hat{A}\,\psi(\mathbf{r})\,d\mathbf{r}$

This gives the average result you'd obtain if you measured $\hat{A}$ on many identically prepared systems. For example, $\langle \hat{x} \rangle$ gives the average position, $\langle \hat{p} \rangle$ gives the average momentum, and $\langle \hat{H} \rangle$ gives the average energy (which for a stationary state is just $E$ ).

The time-independence of expectation values in stationary states is precisely why they're called "stationary." The system isn't frozen; the wave function still oscillates via $e^{-iEt/\hbar}$ . But that phase cancels out in $|\Psi|^2$ and in all expectation values.

Time Evolution of Quantum Systems

Time-Dependent Solutions

When a system isn't in a single stationary state, its wave function is a superposition:

$\Psi(\mathbf{r}, t) = \sum_n c_n\,\psi_n(\mathbf{r})\,e^{-iE_n t/\hbar}$

Each term oscillates at a different frequency $\omega_n = E_n/\hbar$ . Because these frequencies differ, the interference pattern between terms changes with time, and the probability density $|\Psi|^2$ now does depend on $t$ .

This produces quantum beats: oscillations in the probability density at frequencies proportional to the energy differences $|E_m - E_n|/\hbar$ between the states in the superposition. For a two-state superposition, the probability density oscillates back and forth at a single beat frequency.

The coefficients $c_n$ are set by the initial state $\Psi(\mathbf{r}, 0)$ and don't change in time (for a time-independent Hamiltonian). Their squared moduli $|c_n|^2$ give the probability of measuring energy $E_n$ .

Interaction with External Fields

When a quantum system interacts with an external field (an oscillating electric field from a laser, a magnetic field in NMR), the Hamiltonian becomes time-dependent:

$\hat{H}(t) = \hat{H}_0 + \hat{V}(t)$

where $\hat{H}_0$ is the unperturbed Hamiltonian and $\hat{V}(t)$ describes the interaction. Now the coefficients $c_n(t)$ themselves change with time, meaning the system can transition between stationary states of $\hat{H}_0$ .

This is the mechanism behind:

Absorption: The field drives the system from a lower to a higher energy state.
Stimulated emission: The field drives the system from a higher to a lower energy state.
Rabi oscillations: Under a strong resonant driving field, the population oscillates coherently between two states at the Rabi frequency.

Quantum Dynamics

Beyond transitions driven by external fields, the time-dependent Schrödinger equation governs all quantum dynamical processes:

Wave packet motion: A localized wave packet (a superposition of many energy eigenstates) moves through space, and its center follows the classical trajectory in the correspondence limit. Over time, the packet spreads because different momentum components travel at different speeds.
Quantum tunneling: A particle encountering a potential barrier has a nonzero probability of appearing on the other side, even when its energy is below the barrier height. The tunneling probability depends exponentially on the barrier width and height.
Scattering: Incoming wave packets interact with a potential and split into reflected and transmitted components.

For systems too complex to solve analytically, numerical methods are essential. The split-operator method separates the kinetic and potential parts of the propagator, while finite-difference time-domain methods discretize space and time directly. These tools are used to simulate molecular dynamics, reaction dynamics, and quantum control in real chemical systems.

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