Vector Potential (Homework 9, Physics 325)

2. Vector potential in Quantum Mechanics

a) Verify that the vector potential A in the Schrodinger equation for the wave-function $ψ(\mathbf{r})$ in the presence of a magnetic field can be eliminated by a “gauge transformation,” i.e., $ψ(\mathbf{r}) = ψ′(r)exp[i(q/\hbar) \int \mathbf{A}(\mathbf{r}) · d\mathbf{r}$ where $ψ'(\mathbf{r})$ is the solution to the Schrodinger equation in the absence of a magnetic field $\mathbf{B} = ∇ × \mathbf{A}$.

b) Assuming that $<ψ_n|∇_R|ψ_n> = −i(q/\hbar)\int \mathbf{A}(R)$, verify the observation that the Berry phase $γ_n = q \Phi/\hbar$ where \Phi= \int \mathbf{B}  d\mathbf{S}$ is the magnetic flux through a path for a complete cycle.


Assuming no electric field and the presence of a magnetic vector potential $A ⃗(R ⃗ )$ with $B ⃗=∇×A ⃗ $ the Hamiltonian of the charged particle q is

$H=1/2m*(p-q A)^2+V=1/2m (ℏ/i ∇-q A)^2+V$

We take the solution ψ of the above Sch equation

$[1/2m (ℏ/i ∇-q A)^2+V]ψ=iℏ* dψ/d t$

to be a Gage transformation:

$ψ=e^{i g(r)} *ψ’$ where $g(r)=q/ℏ ∫_O^r ⃗ A ⃗(r ⃗ )d R ⃗ $

(Observation :The integral (between an arbitrary point O and final $r ⃗$) need to not depend on path between ending points and therefore it is necessary $∇×A ⃗=0 (=B ⃗)$)

Begin by writing

$∇ψ=i∇g(e^{i g} ψ’ )+e^{i g} ∇ψ’$

$(ℏ/i ∇-q A)ψ=ℏ∇g(e^{i g} ψ’ )+ℏ/i e^{i g} ∇ψ’-qAψ=$

$=ℏ(q/ℏ A)ψ+ℏ/i e^{i g} ∇ψ’-qAψ=ℏ/i e^{i g} ∇ψ’$

And therefore

$(ℏ/i ∇-q A)^2 ψ=-ℏ^2 e^{i g} ∇^2 ψ’$

Back into initial Sch equation:

$-ℏ^2/2m e^{i g} ∇^2 ψ’+V(e^{i g} ψ’)=iℏ*e^{i g} (dψ’)/d t$


$- ℏ^2/2m ∇^2 ψ’+Vψ’=iℏ (dψ’)/d t$

which is the Sch equation for $ψ’$ in the absence of the magnetic field.


The Berry phase is

$γ=i∮ <ψ_n |∇ψ_n>d R=i∬ ∇× <ψ_n |∇ψ_n>d S=$

$=q/ℏ ∬ (∇×A ⃗ )d S=q/ℏ ∬BdS=qΦ/ℏ$


$<ψ_n |∇ψ_n>=-i q/ℏ A ⃗(R ⃗)$