Adiabatic Approximation

Created in January 01, 2025

2025   ·   QuantumMechanics

Adiabatic Theorem

Adiabatic Process

The aidbatic process is defined as that with external condition changing slowly. There are two characteristic times: \(T_i\) is the “internal” time, representing the motion of system itself (such as the period of a single pendulum); \(T_e\) is the external time, representing the time needed for system parameters to change significantly (such as the motion of the fixed point of a pendulum). Adiabatic process requires \(T_e >> T_i\)

The basic method to anylyze an adiabatic process is condidering external parameters as constant, only variable at the last of calculation. It can be expressed as the adiabatictheorem below.
Assume Hamiltonian change from initial value \(H^i\) to \(H^f\) slowly. If a particle under the \(n\) order eigen state of \(H^i\), it will evolve to the \(n\) order eigen state of \(H^f\) (following Schrodinger Equation).

Proof of Adiabatic Theorem

We know the solution to TDSE under time-independent Hamiltonian is \begin{equation} \label{10.12} \Psi(t) = \sum_n c_n(t)\psi_n(t)e^{i\theta_n(t)} \end{equation} where \begin{equation} \theta_n(t) = -\frac{1}{\hbar}\int_0^t E_n(t’)dt’. \end{equation} Substitute equation\eqref{10.12} into TDSE, we obtain \begin{equation} \sum_n \dot c_n \psi_n e^{i\theta_n} = -\sum_n c_n \dot\psi_n e^{i\theta_n} \end{equation} Take the inner product with \(\psi_m\), \begin{equation} \sum_n \dot c_n \delta_{mn} e^{i\theta_n} = -\sum_n c_n\langle \psi_m | \dot\psi_n \rangle e^{i\theta_n} \end{equation} That is \begin{equation} \label{10.16} \dot c_m(t) = -\sum_n c_n \langle \psi_m |\dot\psi_n \rangle e^{i(\theta_n- \theta_m)}. \end{equation} Now take the differential of the eigen equation of \(H\): \begin{equation} \dot H \psi_n+H_n \dot\psi_n=\dot E_n \psi_n +E_n \dot\psi_n \end{equation} Take the inner product with \(\psi_m\) again and use the Hermitian of \(H\): \begin{equation} \langle \psi_m |\dot H |\psi_n\rangle = (E_n-E_m)\langle\psi_m | \dot\psi_n\rangle \end{equation} Substitute this equation into \eqref{10.16}, and do the adiabatic approximation (\(\dot H\) is very small) to ignore the second term, we can obatin: \begin{equation} \dot c_m(t) = -c_m \langle\psi_m |\dot\psi_m\rangle, \end{equation} with the solution \begin{equation} c_m(t) = c_m(0) e^{i\gamma_m (t)} \end{equation} where \(\gamma_m(t) = i\int_0^t \langle \psi_m(t') | \frac{\partial}{\partial t'}\psi_m(t') \rangle dt'\). Specially, if the particle is under the \(n\)th eigen state (\(c_n(0)=1\)), then \begin{equation} \label{psi} \Psi_n(t)=e^{i\theta_n(t) t} e^{i\gamma_n(t)} \psi_n(t) \end{equation} Thus it is still under the \(n\)th eigen state, with a phase factor added. (QED)

Berry Phase

Uncomplete System

For a system that moves (or some parameters vary) along a closed curve but does not go back to the initial state, we call it an uncomplete system.

Geometric Phase

In equation\eqref{psi}, \begin{equation} \theta_n(t) = -\frac{1}{\hbar}\int_0^t E_n(t’)dt’ \end{equation} is the dynamic phase, while \begin{equation} \gamma_n(t) = i\int_0^t \langle \psi_n(t’) |\frac{\partial}{\partial t’}\psi_n(t’)\rangle dt’ \end{equation} is called geometric phase. If there are \(N\) parameters varying: \(R_1(t), R_2(t), \cdots\); in this case, \begin{equation} \frac{\partial \psi_n}{\partial t} = \sum_n \frac{\partial\psi_n}{\partial R_n} \frac{dR_n}{dt}=(\nabla_R \psi_n)\cdot \frac{d\vec R}{dt}, \end{equation} where \(\vec R = (R_1,R_2,...,R_N)\). Now we have \begin{equation} \gamma_n(t) = i\int_{\vec R_i}^{\vec R_f} \langle \psi_n | \nabla_R \psi_n\rangle \cdot d\vec R, \end{equation} If the Hamiltonian return to the initial form, then the final geometric phase is \begin{equation} \label{10.45} \gamma_n(T)=i \oint \langle \psi_n |\nabla_R \psi_n\rangle \cdot d\vec R \end{equation}which is a closed integral in parameter space normally non-zero. \(\gamma_n(T)\) is called the Berry phase, and it only depends on the path. In three dimension parameter space, we can write equation\eqref{10.45} into a surface integral: \begin{equation} \gamma_n(T) = i\int [\nabla_R \times \langle\psi_n |\nabla_R \psi_n\rangle ] \cdot d\vec a. \end{equation}

Aharonov-Bohm Effect (AB Effect)

Consider a particle moving along a ring with radious \(b\), and a long soleenoid with current \(I\) and radious \(a\) is located along the axis. Now the EM field outside the solenoid is zero but vector potential is not. Actually, \begin{equation} \vec A=\frac{\Phi}{2\pi r}\hat\phi, (r>a) \end{equation} where \(\Phi = \pi a^2B\) is the magnetic flux in the solenoid. The Hamiltonian can be written into \begin{equation} H=\frac{1}{2m}[-\hbar^2\nabla^2+q^2A^2+2i\hbar q\vec A\cdot\nabla]. \end{equation} As the wave function is independent of \(\theta\) and \(r\), \(\nabla \to \frac{\hat\phi}{b}\frac{d}{d\phi}\), the Schrodinger Equation becomes \begin{equation} \frac{d^2 \psi}{d\phi^2}-2i\beta \frac{d\psi}{d\phi}+\epsilon\psi = 0, \end{equation} where \(\beta=\frac{q\phi}{2\pi\hbar}\) and \(\epsilon=\frac{2mb^2E}{\hbar^2}-\beta\). The solution is in the following form: \begin{equation} \psi = Ae^{i\lambda\phi}, \end{equation} where \begin{equation} \lambda = \beta \pm \sqrt{\beta^2+\epsilon}=\beta \pm \frac{b}{\hbar}\sqrt{2mE}. \end{equation} The periodic boundary condition requires \(\lambda = n\), then: \begin{equation} E_n=\frac{\hbar^2}{2mb^2}(n-\frac{q\Phi}{2\pi\hbar})^2, \quad (n=0,\pm 1,\pm 2,\cdots) \end{equation} We can see the double degeneracy is splitted, and the allowed value of energy is dependent on the internal field of solenoid, though the field is zero at the particle.