QuantumPhysics.dvi
wang
(Wang)
#1
7.2 Constant Magnetic fields
An important special case is when the electric and magnetic fields areconstant in time, and
uniform in space, so that
A(r,t) =
1
2
B×r Φ(r,t) =−E·r (7.13)
Here, we have chosen a particular convenient gauge. The Hamiltonian is then time-independent,
and we may specialize to the eigenfunctionsψE(r) at fixed energyE, which now obey the
time-independent Schr ̈odinger equation,
1
2 m
(
−i ̄h∇r−
1
2
eB×r
) 2
ψE(r)−eE·rψE(r) =EψE(r) (7.14)
This system is akin to a harmonic oscillator problem. Instead of tryingto solve the Schr ̈odinger
equation explicitly (in terms of Hermite polynomials etc), we shall fully exploit its relation
with the harmonic oscillator and solve for the spectrum using operator methods.
In fact, we will setE= 0, concentrating on the purely magnetic problem. Choosing the
z-axis to coincide with the direction ofB, the problem may be reduced to a 2-dimensional
one, since motion along thez-direction is insensitive to the magnetic field. To make the
problem even more interesting, we add a harmonic oscillator with frequencyω. Thus, the
effective Hamiltonian of the system is now,
H=
1
2 m
(
px+
1
2
eBy
) 2
+
1
2 m
(
py−
1
2
eBx
) 2
+
1
2
mω^2 (x^2 +y^2 ) (7.15)
Combining terms of the formx^2 +y^2 , we may recastHas follows,
H=
1
2 m
(p^2 x+p^2 y)−
eB
2 m
(xpy−ypx) +
1
2
mω^2 B(x^2 +y^2 ) (7.16)
where we have defined the frequencyωBby,
ωB^2 =ω^2 +
e^2 B^2
4 m^2
(7.17)
We recognize the middle term inHas the orbital angular momentum operatorLz. Clearly,
this term commutes withHand may be diagonalized simultaneously withH.
7.2.1 Map onto harmonic oscillators
To solve this system, we introduce the following harmonic oscillator combinations,
a 1 =
1
√
2 m ̄hωB
(ipx+mωBx)
a 2 =
1
√
2 m ̄hωB
(ipy+mωBy) (7.18)