Approximation Methods
Most quantum systems cannot be solved exactly. The hydrogen atom is solvable; helium (just one more electron) is not. Real molecules, solids, and field theories all require approximation methods. The art of physics is knowing which approximation to use and understanding its limits.
The Toolkit
- Perturbation theory: When the problem is "almost" solvable
- Variational method: When you can guess the shape of the solution
- WKB approximation: When quantum effects are "small"
What is a perturbation?
A perturbation is a small disturbance or modification added to a system you already understand. Imagine you know exactly how a guitar string vibrates, and then you tape a tiny piece of clay to it. The clay is the "perturbation," and perturbation theory lets you calculate how it shifts the vibration frequencies without solving the whole problem from scratch.
Perturbation Theory
The workhorse of quantum physics. Start with a problem you can solve, then add a small correction.
The Setup
Reminder: the Hamiltonian
The Hamiltonian $\hat{H}$ is the operator representing the total energy of a quantum system. Solving the equation $\hat{H}|\psi\rangle = E|\psi\rangle$ gives us the allowed energy levels $E$ and their corresponding states $|\psi\rangle$. In perturbation theory, we split this operator into a part we can solve and a small correction.
Suppose the Hamiltonian can be written as:
$$\hat{H} = \hat{H}_0 + \lambda \hat{H}'$$where $\hat{H}_0$ is the "unperturbed" Hamiltonian (exactly solvable), $\hat{H}'$ is the perturbation, and $\lambda$ is a small parameter (often set to 1 at the end). We know the solutions to $\hat{H}_0$:
$$\hat{H}_0|n^{(0)}\rangle = E_n^{(0)}|n^{(0)}\rangle$$The Expansion
Power series expansion
A power series is a way of expressing a quantity as a sum of increasingly small corrections: a zeroth-order term, plus a first-order correction, plus a second-order correction, and so on. It is like estimating a distance by first eyeballing it, then refining with a ruler, then with a micrometer. Each additional term gives a more precise answer.
Expand energies and states in powers of $\lambda$:
$$E_n = E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + \cdots$$ $$|n\rangle = |n^{(0)}\rangle + \lambda|n^{(1)}\rangle + \lambda^2|n^{(2)}\rangle + \cdots$$First-Order Energy Shift
$$E_n^{(1)} = \langle n^{(0)}|\hat{H}'|n^{(0)}\rangle$$The expectation value of the perturbation in the unperturbed state.
Expectation value
The expectation value $\langle n|\hat{O}|n\rangle$ is the average result you would obtain if you measured the quantity represented by operator $\hat{O}$ many times on a system in state $|n\rangle$. It is the quantum-mechanical version of a weighted average, where the wavefunction determines the weights.
This is intuitive: the energy shift is the "average" of the perturbation.
Second-Order Energy Shift
$$E_n^{(2)} = \sum_{m \neq n} \frac{|\langle m^{(0)}|\hat{H}'|n^{(0)}\rangle|^2}{E_n^{(0)} - E_m^{(0)}}$$Key observations:
- Involves a sum over all other states
- Nearby states (small energy denominator) contribute most
- For the ground state, all terms are negative (energy decreases)
- The perturbation "mixes in" other states
When Perturbation Theory Fails
The method breaks down when:
- $\hat{H}'$ is not "small" compared to $\hat{H}_0$
- States are degenerate ($E_n^{(0)} = E_m^{(0)}$ makes the denominator zero)
- The series doesn't converge
Degenerate perturbation theory handles the second case, but sometimes perturbation theory simply isn't the right tool.
Time-Dependent Perturbation Theory
When the perturbation varies in time, $\hat{H}' = \hat{H}'(t)$, we calculate transition probabilities between states. If the perturbation is turned on at $t=0$, the probability of transitioning from state $|i\rangle$ to $|f\rangle$ is:
$$P_{i \to f}(t) = \frac{1}{\hbar^2}\left|\int_0^t \langle f|\hat{H}'(t')|i\rangle e^{i\omega_{fi}t'} dt'\right|^2$$where $\omega_{fi} = (E_f - E_i)/\hbar$.
Fermi's Golden Rule
For a constant perturbation turned on suddenly:
$$\Gamma_{i \to f} = \frac{2\pi}{\hbar}|\langle f|\hat{H}'|i\rangle|^2 \rho(E_f)$$where $\rho(E_f)$ is the density of final states. This gives the transition rate per unit time.
Density of states
The density of states $\rho(E)$ counts how many quantum states are available per unit energy interval near energy $E$. If states are closely packed together, $\rho$ is large and transitions are more likely. If states are sparse, transitions are rarer. Think of it as measuring how crowded the "energy ladder" is at a given rung.
The Variational Method
A completely different approach: guess a solution and optimize it.
Trial wavefunction
A trial wavefunction is an educated guess for what the true wavefunction looks like, written with one or more adjustable parameters. You do not need to guess perfectly. The variational method guarantees that optimizing your guess will always give an energy that is at or above the true ground-state energy, so even a rough guess can produce a useful result.
The Variational Principle
For any normalized trial wavefunction $|\psi_{\text{trial}}\rangle$:
Variational Inequality
$$\langle\psi_{\text{trial}}|\hat{H}|\psi_{\text{trial}}\rangle \geq E_0$$The expectation value of energy is always greater than or equal to the true ground state energy.
Equality holds only when $|\psi_{\text{trial}}\rangle$ is the true ground state.
The Method
- Choose a trial wavefunction with adjustable parameters: $|\psi(\alpha, \beta, \ldots)\rangle$
- Calculate the energy expectation value: $E(\alpha, \beta, \ldots) = \langle\psi|\hat{H}|\psi\rangle$
- Minimize with respect to all parameters: $\partial E/\partial\alpha = 0$, etc.
- The minimum gives the best approximation to $E_0$
Example: Helium Atom
The helium Hamiltonian includes electron-electron repulsion, making it unsolvable. A simple trial function treats each electron as seeing an effective nuclear charge $Z_{\text{eff}}$:
$$\psi_{\text{trial}}(r_1, r_2) = \frac{Z_{\text{eff}}^3}{\pi a_0^3} e^{-Z_{\text{eff}}(r_1 + r_2)/a_0}$$The Bohr radius $a_0$
The Bohr radius ($a_0 \approx 0.053$ nm) is the characteristic size of a hydrogen atom. It sets the natural length scale for atomic physics. In the formula above, the exponential decay $e^{-r/a_0}$ means the electron is most likely found within a distance of about $a_0$ from the nucleus.
Optimizing gives $Z_{\text{eff}} \approx 1.69$ (instead of 2), accounting for electron screening. The predicted energy is within 2% of experiment, not bad for a one-parameter approximation.
Strengths of the Variational Method
- Always gives an upper bound on the ground state energy
- Works even when perturbation theory fails
- Physical intuition guides the choice of trial function
- Can be systematically improved by adding more parameters
WKB Approximation
Named for Wentzel, Kramers, and Brillouin. A semiclassical method valid when the wavefunction varies slowly, when quantum effects are "gentle."
What does "semiclassical" mean?
A semiclassical method sits halfway between fully classical and fully quantum physics. It uses classical concepts (like particle trajectories and momentum) but adds quantum corrections on top. Semiclassical methods work best when a system is "almost classical," meaning the quantum wavelength of the particle is much smaller than the scale over which the potential changes.
The Idea
What is an ansatz?
An ansatz (from German, meaning "starting point") is an assumed mathematical form for the solution. Rather than deriving the answer from scratch, you propose a template and then check whether it works or determine the conditions under which it holds. It is a common and powerful strategy throughout physics.
For a free particle with momentum $p$, the wavefunction is $e^{ipx/\hbar}$. For a particle in a potential, the momentum varies with position. The WKB ansatz is:
$$\psi(x) \approx \frac{C}{\sqrt{p(x)}} \exp\left(\pm\frac{i}{\hbar}\int p(x)\, dx\right)$$where the local momentum is:
$$p(x) = \sqrt{2m(E - V(x))}$$Validity Condition
The de Broglie wavelength
Every moving particle has an associated wavelength $\lambda = h/p$, where $p$ is its momentum and $h$ is Planck's constant. A fast, heavy particle has a tiny wavelength (behaves classically), while a slow, light particle has a large wavelength (behaves quantum mechanically). WKB works when this wavelength is small compared to the scale of the potential.
WKB is valid when the de Broglie wavelength changes slowly:
$$\left|\frac{d\lambda}{dx}\right| \ll 1$$Classical turning points
A turning point is the location where a classical particle would stop and reverse direction because it has run out of kinetic energy. At these points the particle's momentum drops to zero. Think of a ball rolling up a hill: the point where it stops and rolls back is the turning point. In quantum mechanics, the wavefunction changes character at turning points, switching from oscillating to exponentially decaying.
This fails at classical turning points where $E = V(x)$ and $p(x) = 0$.
The Bohr-Sommerfeld Quantization
Matching the wavefunction at turning points gives a quantization condition:
WKB Quantization
$$\oint p(x)\, dx = \left(n + \frac{1}{2}\right)h$$The integral is over one complete classical orbit.
For the harmonic oscillator, this gives exactly $E_n = \hbar\omega(n + 1/2)$. For other potentials, it's an approximation that improves at large $n$.
Tunneling
What is quantum tunneling?
In classical physics, a ball cannot pass through a hill if it does not have enough energy to go over the top. In quantum mechanics, however, a particle has a nonzero probability of appearing on the other side of an energy barrier even when it lacks the energy to cross classically. This phenomenon is called tunneling. It is responsible for radioactive decay, the operation of flash memory, and many chemical reactions.
WKB beautifully explains quantum tunneling. In classically forbidden regions where $E < V(x)$:
$$\psi(x) \propto \exp\left(-\frac{1}{\hbar}\int |p(x)|\, dx\right)$$The transmission probability through a barrier is approximately:
$$T \approx \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2} |p(x)|\, dx\right)$$where $x_1$ and $x_2$ are the turning points where $E = V(x)$.
Applications of Tunneling
- Alpha decay: The alpha particle tunnels through the nuclear potential barrier
- Scanning tunneling microscope: Electrons tunnel between tip and surface
- Quantum field theory: Instantons describe tunneling between vacua
Choosing the Right Method
| Method | Best When | Limitations |
|---|---|---|
| Perturbation theory | Small deviations from solvable problem | Fails for strong perturbations, degeneracy |
| Variational method | Ground state with good physical intuition | Result depends on trial function quality |
| WKB | Smooth potentials, high quantum numbers | Fails near turning points, rapid variation |
In practice, physicists often combine methods: use variational for the ground state, perturbation for excited states, and WKB for semiclassical limits.
- Most quantum problems require approximation. Exact solutions are rare
- Perturbation theory expands around a known solution; first-order correction is the expectation value
- The variational principle guarantees any trial function gives an upper bound on energy
- WKB connects quantum and classical mechanics; valid when the wavelength varies slowly
- Tunneling (classically forbidden transmission) emerges naturally from WKB
- Fermi's Golden Rule gives transition rates for time-dependent perturbations
- Choosing the right method is an art. Each has its domain of validity