Variational Methods, WKB Semiclassical Approximation & Dynamical Approximations

1. The Rayleigh-Ritz Variational Theorem

The variational method provides a rigorous technique to establish an upper bound on the ground state energy $E_0$ of any quantum system described by a Hamiltonian $\hat{H}$. Let $\{|n\rangle\}$ be the complete unknown set of exact orthonormal energy eigenstates with true eigenvalues $E_0 \le E_1 \le E_2 \le \dots$.

Consider an arbitrary normalized trial state ket $|\psi_{\text{trial}}\rangle \in \mathcal{H}$. Expanding in the exact basis:

$$|\psi_{\text{trial}}\rangle = \sum_n c_n |n\rangle, \quad \sum_n |c_n|^2 = 1$$

The energy expectation value (the Rayleigh quotient) evaluates to:

$$\langle \hat{H} \rangle_{\text{trial}} = \langle \psi_{\text{trial}} | \hat{H} | \psi_{\text{trial}} \rangle = \sum_n |c_n|^2 E_n$$

Subtracting the true ground-state energy $E_0$:

$$\langle \hat{H} \rangle_{\text{trial}} - E_0 = \sum_n |c_n|^2 (E_n - E_0) \ge 0$$

Since $E_n \ge E_0$ and $|c_n|^2 \ge 0$ for all $n$, we obtain the fundamental Variational Inequality:

$$\langle \hat{H} \rangle_{\text{trial}} = \frac{\langle \psi_{\text{trial}} | \hat{H} | \psi_{\text{trial}} \rangle}{\langle \psi_{\text{trial}} | \psi_{\text{trial}} \rangle} \ge E_0$$

Equality holds if and only if $|\psi_{\text{trial}}\rangle = |0\rangle$.

2. Optimization with Variational Parameters

In practice, one selects an educated ansatz $\psi(x; \alpha_1, \alpha_2, \dots, \alpha_k)$ containing adjustable variational parameters $\{\alpha_i\}$. The best approximation to the ground state is found by minimizing the Rayleigh quotient:

$$\frac{\partial \langle \hat{H} \rangle}{\partial \alpha_i} = 0 \quad (i = 1, \dots, k)$$

The minimum value $\langle \hat{H} \rangle_{\min}$ serves as an upper bound to the true ground state energy $E_0$.

1. The Ground State of the Helium Atom

The Helium atom ($Z=2$) consists of two electrons orbiting a nucleus of charge $+2e$. Neglecting nuclear recoil:

$$\hat{H} = -\frac{\hbar^2}{2m}\nabla_1^2 - \frac{Ze^2}{4\pi\varepsilon_0 r_1} - \frac{\hbar^2}{2m}\nabla_2^2 - \frac{Ze^2}{4\pi\varepsilon_0 r_2} + \frac{e^2}{4\pi\varepsilon_0 |\vec{r}_1 - \vec{r}_2|}$$

Without the electron-electron repulsion, the ground state would be two independent $1s$ electrons with energy $2 \times (-Z^2 \times 13.6\text{ eV}) = -108.8\text{ eV}$. However, the experimental ground state energy is $-79.0\text{ eV}$.

2. Variational Screening Parameter $Z_{\text{eff}}$

Because each electron partially screens the nuclear charge felt by the other, we choose a hydrogenic trial wavefunction with an effective nuclear charge $Z^*$ as the variational parameter:

$$\psi(\vec{r}_1, \vec{r}_2; Z^*) = \frac{Z^{*3}}{\pi a_0^3} \exp\left(-\frac{Z^*(r_1 + r_2)}{a_0}\right)$$

Rewriting the Hamiltonian in terms of the effective Hamiltonian:

$$\hat{H} = \hat{H}_0(Z^*) + (Z^* - Z)\frac{e^2}{4\pi\varepsilon_0}\left(\frac{1}{r_1} + \frac{1}{r_2}\right) + \frac{e^2}{4\pi\varepsilon_0 r_{12}}$$

Evaluating the expectation values:

$$\langle \hat{H}_0(Z^*) \rangle = 2 \left(-Z^{*2} E_R\right) = -2 Z^{*2} E_R \quad (E_R = 13.6\text{ eV})$$ $$\left\langle \frac{1}{r_1} + \frac{1}{r_2} \right\rangle = 2\frac{Z^*}{a_0} \implies \langle V_{\text{screen}} \rangle = 4(Z^* - Z)Z^* E_R$$ $$\left\langle \frac{e^2}{4\pi\varepsilon_0 r_{12}} \right\rangle = \frac{5}{8}Z^* \left(\frac{e^2}{4\pi\varepsilon_0 a_0}\right) = \frac{5}{4}Z^* E_R$$

Summing all terms yields the energy expectation value as a function of $Z^*$:

$$E(Z^*) = \left[ -2Z^{*2} + 4(Z^* - Z)Z^* + \frac{5}{4}Z^* \right] E_R = \left[ 2Z^{*2} - 4ZZ^* + \frac{5}{4}Z^* \right] E_R$$

Minimizing with respect to $Z^*$:

$$\frac{dE}{dZ^*} = \left( 4Z^* - 4Z + \frac{5}{4} \right) E_R = 0 \implies Z^* = Z - \frac{5}{16}$$

For Helium ($Z = 2$):

$$Z^* = 2 - \frac{5}{16} = \frac{27}{16} = 1.6875$$

Each electron screens the nucleus by approximately $0.3125$ electronic charges. The variational ground state energy is:

$$E(Z^*) = -2 \left(Z - \frac{5}{16}\right)^2 E_R = -2 \left(\frac{27}{16}\right)^2 (13.6\text{ eV}) = -\frac{729}{128}(13.6\text{ eV}) \approx -77.46\text{ eV}$$

This is within $1.9\%$ of the true experimental value ($-79.0\text{ eV}$).

1. The Ritz Linear Combination of Basis Functions

When a system cannot be easily modeled by a simple parameterized function, one expands the trial wavefunction as a linear combination of $N$ known linearly independent basis functions $\{\phi_j\}$:

$$\psi_{\text{trial}} = \sum_{j=1}^N c_j \phi_j$$

The energy expectation value is:

$$E = \frac{\sum_{i=1}^N \sum_{j=1}^N c_i^* c_j H_{ij}}{\sum_{i=1}^N \sum_{j=1}^N c_i^* c_j S_{ij}}$$

where $H_{ij} = \langle \phi_i | \hat{H} | \phi_j \rangle$ is the Hamiltonian matrix element and $S_{ij} = \langle \phi_i | \phi_j \rangle$ is the overlap matrix element.

2. The Generalized Secular Equation

Multiplying by the denominator $\sum_{i,j} c_i^* c_j S_{ij}$ and differentiating with respect to $c_k^*$:

$$\frac{\partial}{\partial c_k^*} \left[ \sum_{i,j} c_i^* c_j (H_{ij} - E S_{ij}) \right] = 0 \implies \sum_{j=1}^N (H_{kj} - E S_{kj}) c_j = 0$$

This yields a system of linear homogeneous equations, with non-trivial solutions given by the generalized secular determinant:

$$\det(\mathbf{H} - E \mathbf{S}) = 0$$

The $N$ real roots $E_1 \le E_2 \le \dots \le E_N$ provide rigorous upper bounds not only for the ground state ($E_1 \ge E_0$), but also for the first $N-1$ excited states (the Hylleraas-Undheim-MacDonald Theorem):

$$E_k \ge E_k^{\text{true}} \quad (k = 1, \dots, N)$$

1. The Wentzel-Kramers-Brillouin (WKB) Semiclassical Expansion

The WKB method applies to systems where the potential energy $V(x)$ varies slowly compared to the local de Broglie wavelength of the particle. Starting from the 1D time-independent Schrödinger equation:

$$\frac{d^2\psi}{dx^2} + \frac{2m}{\hbar^2}(E - V(x))\psi(x) = 0 \implies \frac{d^2\psi}{dx^2} + \frac{p^2(x)}{\hbar^2}\psi(x) = 0$$

where $p(x) = \sqrt{2m(E - V(x))}$ is the classical local momentum. Writing the wave function in terms of an action phase $S(x)$:

$$\psi(x) = \exp\left(\frac{i}{\hbar}S(x)\right)$$

Substituting into the Schrödinger equation:

$$\frac{d\psi}{dx} = \frac{i}{\hbar}S'\psi, \quad \frac{d^2\psi}{dx^2} = \left(\frac{i}{\hbar}S'' - \frac{1}{\hbar^2}(S')^2\right)\psi$$ $$\implies -(S')^2 + i\hbar S'' + p^2(x) = 0$$

Expanding $S(x)$ as a power series in Planck's constant $\hbar$:

$$S(x) = S_0(x) + \hbar S_1(x) + \hbar^2 S_2(x) + \dots$$

Collecting orders of $\hbar$:

$$\mathcal{O}(\hbar^0): \quad -(S_0')^2 + p^2(x) = 0 \implies S_0'(x) = \pm p(x) \implies S_0(x) = \pm \int p(x)\,dx$$ $$\mathcal{O}(\hbar^1): \quad -2 S_0' S_1' + i S_0'' = 0 \implies S_1'(x) = \frac{i S_0''}{2 S_0'} = \frac{i p'(x)}{2 p(x)}$$ $$S_1(x) = \frac{i}{2}\ln p(x) \implies \exp(i S_1(x)) = \exp\left(-\frac{1}{2}\ln p(x)\right) = \frac{1}{\sqrt{p(x)}}$$

2. The Semiclassical Wavefunctions and Validity Criterion

To first order in $\hbar$, the WKB wavefunction is:

1. Classically Allowed Region ($E > V(x)$, $p(x) \in \mathbb{R}$): $$\psi(x) \approx \frac{A}{\sqrt{p(x)}}\exp\left(\frac{i}{\hbar}\int p(x)\,dx\right) + \frac{B}{\sqrt{p(x)}}\exp\left(-\frac{i}{\hbar}\int p(x)\,dx\right)$$ The probability density $|\psi(x)|^2 \propto \frac{1}{p(x)} \propto \frac{1}{v(x)}$ correctly mirrors the classical dwell time: a particle spends the most time where it travels slowest.
2. Classically Forbidden Region ($E < V(x)$, $p(x) = i|p(x)|$): $$\psi(x) \approx \frac{C}{\sqrt{|p(x)|}}\exp\left(-\frac{1}{\hbar}\int |p(x)|\,dx\right) + \frac{D}{\sqrt{|p(x)|}}\exp\left(+\frac{1}{\hbar}\int |p(x)|\,dx\right)$$

WKB Validity Criterion: The condition $|\hbar S_1'| \ll |S_0'|$ requires:

$$\left| \frac{\hbar p'(x)}{2 p^2(x)} \right| \ll 1 \iff \left| \frac{d}{dx}\left(\frac{\hbar}{p(x)}\right) \right| = \left| \frac{d\bar{\lambda}(x)}{dx} \right| \ll 1$$

The reduced de Broglie wavelength must vary slowly over a distance of one wavelength. This condition fails catastrophically at classical turning points ($E = V(x)$), where $p(x) \to 0$ and $\lambda(x) \to \infty$.

1. Linearization Around Classical Turning Points

Near a turning point $x = x_0$ where $V(x_0) = E$, the potential can be linearized via a Taylor series:

$$V(x) \approx E + V'(x_0)(x - x_0)$$

The Schrödinger equation becomes:

$$\frac{d^2\psi}{dx^2} - \frac{2m V'(x_0)}{\hbar^2}(x - x_0)\psi = 0$$

Defining the dimensionless coordinate $z = \left(\frac{2m V'(x_0)}{\hbar^2}\right)^{1/3}(x - x_0)$, this reduces to the canonical Airy Differential Equation:

$$\frac{d^2\psi}{dz^2} - z\psi = 0$$

Its general solution is a linear combination of the Airy functions $\text{Ai}(z)$ and $\text{Bi}(z)$. For a bound state that decays into the forbidden region $z > 0$, the physically admissible solution is $\text{Ai}(z)$.

2. Asymptotic Matching and Connection Formulae

Using the known asymptotic expansions of the Airy function $\text{Ai}(z)$:

$$\text{Ai}(z) \sim \begin{cases} \frac{1}{2\sqrt{\pi} z^{1/4}}\exp\left(-\frac{2}{3}z^{3/2}\right) & \text{for } z \gg 0 \text{ (forbidden)} \\ \frac{1}{\sqrt{\pi}(-z)^{1/4}}\cos\left(\frac{2}{3}(-z)^{3/2} - \frac{\pi}{4}\right) & \text{for } z \ll 0 \text{ (allowed)} \end{cases}$$

Matching the asymptotic forms of the Airy solution with the WKB expressions on either side of the turning point yields the WKB Connection Formulae:

$$\frac{1}{2\sqrt{|p(x)|}}\exp\left(-\frac{1}{\hbar}\int_{x_0}^x |p(x')|\,dx'\right) \longleftrightarrow \frac{1}{\sqrt{p(x)}}\cos\left(\frac{1}{\hbar}\int_x^{x_0} p(x')\,dx' - \frac{\pi}{4}\right)$$

The phase shift of $-\pi/4$ at a smooth turning point arises directly from tunneling into the linear turning potential.

1. The Bohr-Sommerfeld-Wilson Quantization Condition

Consider a particle bound in a potential well between two classical turning points $x_1$ and $x_2$ ($x_1 < x_2$). Connecting the WKB solution from the left turning point $x_1$ and the right turning point $x_2$:

$$\psi(x) \propto \frac{1}{\sqrt{p(x)}}\cos\left(\frac{1}{\hbar}\int_{x_1}^x p(x')\,dx' - \frac{\pi}{4}\right)$$ $$\psi(x) \propto \frac{1}{\sqrt{p(x)}}\cos\left(\frac{1}{\hbar}\int_x^{x_2} p(x')\,dx' - \frac{\pi}{4}\right)$$

For these two expressions to represent the same single-valued physical wavefunction throughout the allowed region $x_1 < x < x_2$, the sum of their arguments must be an integral multiple of $\pi$:

$$\left(\frac{1}{\hbar}\int_{x_1}^{x_2} p(x')\,dx' - \frac{\pi}{4}\right) + \frac{\pi}{4} = \left(n + \frac{1}{2}\right)\pi$$ $$\int_{x_1}^{x_2} p(x)\,dx = \left(n + \frac{1}{2}\right)\pi\hbar \quad (n = 0, 1, 2, \dots)$$

This is the Bohr-Sommerfeld quantization rule with the Maslov index correction $1/2$ originating from the two smooth $-\pi/4$ turning-point phase losses.

If one boundary is an infinitely rigid wall (e.g. $V(x) = \infty$ for $x \le 0$), the wavefunction vanishes strictly at the boundary ($\psi(0) = 0$), yielding a phase shift of $\pi$ rather than $\pi/4$. The quantization rule becomes $\int p\,dx = (n + 3/4)\pi\hbar$.

2. Quantum Tunneling and Gamow's Theory of Alpha Decay

For a potential barrier extending from $x_1$ to $x_2$ where $V(x) > E$, the WKB transmission probability (tunneling coefficient) through the barrier is:

$$T \approx \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2} |p(x)|\,dx\right) = \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x) - E)}\,dx\right)$$

Applying this to an alpha particle trapped inside a nucleus of radius $R$ attempting to penetrate the Coulomb barrier $V(r) = \frac{2Ze^2}{4\pi\varepsilon_0 r}$ at energy $E$:

$$\gamma = \frac{2}{\hbar}\int_R^{r_c} \sqrt{2m\left(\frac{2Ze^2}{4\pi\varepsilon_0 r} - E\right)}\,dr$$

Carrying out the integration yields the Geiger-Nuttall law:

$$\ln T = -C_1 \frac{Z}{\sqrt{E}} + C_2 \sqrt{ZR} \implies \ln \lambda_{\text{decay}} = A - B \frac{Z}{\sqrt{E}}$$

This explains why a small variation in alpha decay energy produces variations in nuclear half-lives spanning over 20 orders of magnitude.

1. The Quantum Adiabatic Theorem

Consider a Hamiltonian $\hat{H}(t)$ whose parameters vary smoothly and continuously over time $T$. At each instant $t$, there exists an instantaneous orthonormal basis of eigenstates:

$$\hat{H}(t)|\psi_n(t)\rangle = E_n(t)|\psi_n(t)\rangle$$

The Adiabatic Theorem (Born and Fock) states: If a system is initially prepared in the $n$-th discrete, non-degenerate eigenstate $|\psi_n(0)\rangle$ and the Hamiltonian changes sufficiently slowly:

$$\left| \frac{\langle \psi_m(t) | \frac{\partial \hat{H}}{\partial t} | \psi_n(t) \rangle}{E_m(t) - E_n(t)} \right| \ll \frac{|E_m(t) - E_n(t)|}{\hbar} \quad \forall m \ne n$$

then the system will remain in the instantaneous eigenstate $|\psi_n(t)\rangle$ for all $t$, acquiring only phase factors.

2. Berry's Geometric Phase

The evolved state under adiabatic evolution is:

$$|\Psi(t)\rangle = \exp\left(-\frac{i}{\hbar}\int_0^t E_n(t')\,dt'\right) \exp(i\gamma_n(t)) |\psi_n(t)\rangle$$

The first factor is the familiar dynamical phase. Substituting into the time-dependent Schrödinger equation reveals the equation for $\gamma_n(t)$:

$$\dot{\gamma}_n(t) = i \langle \psi_n(t) | \dot{\psi}_n(t) \rangle \implies \gamma_n(C) = i \oint_C \langle \psi_n(\vec{R}) | \nabla_{\vec{R}} \psi_n(\vec{R}) \rangle \cdot d\vec{R}$$

When the parameters trace out a closed circuit $C$ in parameter space, $\gamma_n(C)$ is a non-integrable holonomy depending solely on the geometry and topology of the path—Berry's Phase. Using Stokes' theorem:

$$\gamma_n(C) = \iint_{\Sigma} \vec{\Omega}_n(\vec{R}) \cdot d\vec{\Sigma}$$

where $\vec{\Omega}_n(\vec{R}) = \nabla \times \vec{\mathcal{A}}_n(\vec{R})$ is the Berry curvature.

3. The Sudden Approximation

In the opposite physical extreme, if the Hamiltonian changes abruptly on a time scale $\Delta t \to 0$ much shorter than the characteristic quantum oscillation period $\tau \sim \hbar / \Delta E$:

$$|\psi(t_0 + \Delta t)\rangle \approx |\psi(t_0)\rangle$$

The wavefunction does not have time to respond and remains instantaneously unchanged. The transition probability to find the system in a new energy eigenstate $|m_{\text{new}}\rangle$ is simply the projection:

$$\mathcal{P}_{n \to m} = |\langle m_{\text{new}} | n_{\text{old}} \rangle|^2$$