2.1 Time-dependent and time-independent Schrödinger equations

Time-dependent and Time-independent Schrödinger Equations

The Schrödinger equation governs how quantum systems behave. It comes in two forms: the time-dependent version describes how a system's wavefunction evolves over time, while the time-independent version identifies the stationary states and allowed energy levels. Together, they provide the complete framework for analyzing everything from single particles in wells to electrons in molecules.

Time-dependent vs Time-independent Schrödinger Equations

Describing the evolution and stationary states of quantum systems

The time-dependent Schrödinger equation (TDSE) is the more general form:

$i\hbar \frac{\partial \Psi(x,t)}{\partial t} = \hat{H}\Psi(x,t)$

This is a partial differential equation that tells you how the full wavefunction $\Psi(x,t)$ changes with time. Its solutions are complex-valued functions of both position and time.

The time-independent Schrödinger equation (TISE) is an eigenvalue equation:

$\hat{H}\psi(x) = E\psi(x)$

This version has no time derivative. It asks: for what functions $\psi(x)$ and values $E$ does the Hamiltonian acting on $\psi$ simply return $\psi$ multiplied by a constant? Those functions are the energy eigenstates, and those constants are the energy eigenvalues.

You can derive the TISE from the TDSE through separation of variables. If the potential $V(x)$ doesn't depend on time, you assume the wavefunction factors as:

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

Substituting this into the TDSE and dividing through gives you the TISE for the spatial part $\psi(x)$. The time-dependent factor $e^{-iEt/\hbar}$ contributes only a phase, which is why these states are called "stationary": their probability density $|\Psi(x,t)|^2 = |\psi(x)|^2$ doesn't change in time.

One clarification: the solutions to the TISE are not necessarily real-valued. They can often be chosen to be real for bound states in one dimension, but this isn't guaranteed in general (e.g., states with nonzero angular momentum are complex).

Applications of the time-dependent and time-independent Schrödinger equations

The TDSE is the tool you need whenever a system changes over time:

• 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
• Quantum coherence: a system prepared in a superposition of energy eigenstates evolves with well-defined phase relationships between components
• Wave packet dynamics: tracking how a localized particle spreads and moves through a potential

The TISE is what you solve to find the energy structure of a system:

• Atoms: the hydrogen atom energy levels and orbitals
• Molecules: electronic states of species like $H_2^+$ and $H_2$
• Nanostructures: quantum dots and quantum wells, where confinement determines the allowed energies

The solutions to the TISE also provide the building blocks for solving time-dependent problems, since any general wavefunction can be expanded in the energy eigenbasis.

Solving the Schrödinger Equation for Simple Systems

Setting up and solving the time-independent Schrödinger equation

Solving the TISE follows a consistent procedure:

1. Write down the Hamiltonian for your system by specifying the kinetic energy operator and the potential $V(x)$.
2. Set up the eigenvalue equation $\hat{H}\psi = E\psi$ and write it out as a differential equation.
3. Apply boundary conditions appropriate to the physical situation (e.g., the wavefunction must vanish at the walls of an infinite well, or go to zero at infinity for bound states).
4. Solve the differential equation to find the eigenfunctions $\psi_n(x)$ and eigenvalues $E_n$, where $n$ is a quantum number labeling each state.
5. Normalize the solutions so that $\int |\psi_n(x)|^2 dx = 1$.

The resulting eigenfunctions form a complete orthonormal set. Orthonormality means:

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

where $\delta_{mn}$ is the Kronecker delta (1 if $m = n$, 0 otherwise). Completeness means any well-behaved function in the space can be expanded as a sum of these eigenfunctions.

Analytically solvable quantum systems

Three systems show up constantly because they can be solved exactly and they model real physical situations:

Infinite square well (particle in a box with impenetrable walls):

• Energy eigenvalues: $E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$, where $n = 1, 2, 3, \ldots$, $m$ is the particle mass, and $L$ is the well width
• Eigenfunctions: $\psi_n(x) = \sqrt{\frac{2}{L}} \sin\!\left(\frac{n\pi x}{L}\right)$
• The energy scales as $n^2$, so the spacing between levels increases with $n$

Quantum harmonic oscillator (parabolic potential, models molecular vibrations):

• Energy eigenvalues: $E_n = \left(n + \frac{1}{2}\right) \hbar\omega$, where $n = 0, 1, 2, \ldots$ and $\omega$ is the angular frequency
• Eigenfunctions: $\psi_n(x) = \frac{1}{\sqrt{2^n n!}} \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \exp\!\left(-\frac{m\omega x^2}{2\hbar}\right) H_n\!\left(\sqrt{\frac{m\omega}{\hbar}}\, x\right)$, where $H_n$ are the Hermite polynomials
• The levels are evenly spaced by $\hbar\omega$, and the ground state ($n=0$) has a nonzero zero-point energy of $\frac{1}{2}\hbar\omega$

Hydrogen atom (Coulomb potential):

• Energy eigenvalues: $E_n = -\frac{13.6 \text{ eV}}{n^2}$, where $n = 1, 2, 3, \ldots$
• Eigenfunctions involve products of radial functions (associated Laguerre polynomials) and angular functions (spherical harmonics), characterized by quantum numbers $n, l, m_l$
• The spacing between levels decreases with increasing $n$, converging toward the ionization threshold at $E = 0$

These exactly solvable systems serve as reference points for tackling more complex problems through approximate methods.

The Hamiltonian Operator in the Schrödinger Equation

Components of the Hamiltonian operator

The Hamiltonian operator $\hat{H}$ represents the total energy of the system. It has two parts:

$\hat{H} = \hat{T} + \hat{V} = -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r})$

• The kinetic energy operator $\hat{T} = -\frac{\hbar^2}{2m}\nabla^2$ involves the Laplacian $\nabla^2$, which is the sum of second spatial derivatives. In one dimension, $\nabla^2 = \frac{d^2}{dx^2}$. In three dimensions, it includes derivatives with respect to all three coordinates.
• The potential energy operator $\hat{V}$ depends on the physical setup: a Coulomb potential for atoms, a parabolic potential for oscillators, etc.

For more advanced problems, the Hamiltonian can include additional terms:

• Magnetic fields: the kinetic term is modified using the vector potential $\vec{A}$
• Spin-orbit coupling: an interaction between a particle's orbital motion and its spin, proportional to $(\nabla V \times \hat{p}) \cdot \vec{\sigma}$, where $\vec{\sigma}$ represents the Pauli matrices

The form of the Hamiltonian determines everything about the physics of the system, so writing it down correctly is always the first step.

Properties of the Hamiltonian operator

The Hamiltonian is a Hermitian (self-adjoint) operator. This has two major consequences:

• All energy eigenvalues are real. You'll never get a complex number for a measurable energy. This is physically required since energies are observable quantities.
• Eigenfunctions corresponding to different eigenvalues are orthogonal. Combined with normalization, this gives you the orthonormal basis discussed above.

Hermiticity means the operator satisfies:

$\langle \psi_m | \hat{H} | \psi_n \rangle = \langle \psi_n | \hat{H} | \psi_m \rangle^*$

The completeness of the energy eigenfunctions is what makes them so powerful. Any arbitrary wavefunction in the system's Hilbert space can be written as:

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

where the coefficients $c_n$ are complex numbers found by projection: $c_n = \langle \psi_n | \Psi \rangle$. This expansion is the quantum-mechanical analog of a Fourier series.

With this basis, you can compute expectation values and matrix elements of any observable $\hat{O}$ using:

$\langle \psi_m | \hat{O} | \psi_n \rangle = \int \psi_m^*(x)\, \hat{O}\, \psi_n(x)\, dx$

Energy Eigenvalues and Eigenfunctions

Applying the Schrödinger equation to calculate energy eigenvalues and eigenfunctions

The general workflow for finding eigenvalues and eigenfunctions:

1. Construct the Hamiltonian $\hat{H}$ with the appropriate kinetic and potential terms.
2. Write out $\hat{H}\psi = E\psi$ as an explicit differential equation.
3. For simple potentials (infinite well, harmonic oscillator), solve analytically using techniques like power series expansions or operator methods (raising/lowering operators for the harmonic oscillator).
4. For complex systems (multi-electron atoms, molecules), use approximate methods:
   • Variational method: propose a trial wavefunction with adjustable parameters and minimize $\langle \psi | \hat{H} | \psi \rangle$. This always gives an upper bound on the true ground-state energy.
   • Perturbation theory: start from a system you can solve exactly, then treat the additional complexity as a small perturbation. This yields corrections to both energies and wavefunctions order by order.

Applications of energy eigenvalues and eigenfunctions

Once you have the eigenvalues and eigenfunctions, you can extract all the physics of the system:

• Probability density: $|\psi_n(x)|^2$ tells you where the particle is likely to be found in state $n$
• Expectation values: for any observable $\hat{O}$, the average measured value in state $\psi$ is $\langle \hat{O} \rangle = \langle \psi | \hat{O} | \psi \rangle$
• Transition probabilities: the probability of a transition from state $m$ to state $n$ under a perturbation $\hat{V}'$ is proportional to $|\langle \psi_n | \hat{V}' | \psi_m \rangle|^2$

The energy eigenvalues directly determine the spectroscopic transitions of the system. The frequency of a photon absorbed or emitted during a transition between levels $m$ and $n$ is:

$\nu = \frac{|E_n - E_m|}{h}$

This connects the abstract eigenvalue problem to measurable absorption, emission, and Raman spectra.

The eigenfunctions give you spatial information: atomic and molecular orbitals are just the eigenfunctions (or linear combinations of them) for the relevant Hamiltonian. Visualizing $|\psi|^2$ for different quantum numbers reveals the nodal structure and spatial extent of each state, which in turn governs chemical bonding and reactivity.

In applied contexts, control over energy eigenvalues and eigenfunctions is central to designing quantum-confined structures like quantum dots and quantum wells, where the discrete energy spectrum depends sensitively on the size and shape of the confining potential.