By completing the square we get a term−~^2 ky^2 which cancels with the one already there and we are
left with
1
2 m
(−~^2 φ′′+~^2 k^2 zφ+ (qBzx−~ky)^2 φ) =E′φ
1
2 m
(−~^2 φ′′+q^2 B^2 z(x−~ky
qBz
)^2 φ) = (E′−~^2 kz^2
2 m
)φ≡E′′φ−
~^2
2 m
φ′′+1
2
mq^2 Bz^2
m^2
(x−~y
qBz
)^2 =E′′φ (23)We see that thez-part of the EnergyEz=~
(^2) k (^2) z
2 m is the same as for a free electron. This is because a
constant magnetic field pointing in the z-direction leaves the motion of the particle in the z-direction
unchanged. Comparing eqn.(23) to the equation of a harmonic oscillator
−
~
2 m
φ′′+1
2
mω^2 (x−xo)^2 =Eφ (24)with energiesEn=~ω(n+^12 ), we obtain
En′′=~qBz
m
(n+1
2
)
En′ =~qBz
m
(n+1
2
) +
~^2 k^2 z
2 m
En=~qBz
m
(n+1
2
) +
~^2 k^2 z
2 m
∓gμBBz (25)Where each energy levelEnis split into two levels because of the term∓gμBBz. Where the minus
is for Spin up (parallel toB), which lowers the energy because the Spin is aligned and plus for Spin
down (anti parallel). It should be noted thatωc= qBmz is also the classical angular velocity of an
electron in a magnetic field. Rewriting eqn.(25) by using the cyclotron frequencyωcandμB= 2 qm~,
we end up with
En=~^2 k^2 z
2 m
+~ωc(n+1
2
)∓
g
2
ωcEn=~^2 k^2 z
2 m
+~ωc(n+1 ∓g
2) (26)
3.1.4 Dissipation
Quantum coherence is maintained until the decoherence time. This depends on the strength of the
coupling of the quantum system to other degrees of freedom. As an example: Schrödinger’s cat. The
decoherence time is very short, because there is a lot of coupling with other, lots of degrees of freedom.
Therefore a cat, which is dead and alive at the same time, can’t be investigated.
In solid state physics energy is exchanged between the electrons and the phonons. There are Hamil-
toniansHfor the electrons (He), the phonons (Hph) and the coupling between both (He−ph). The
energy is conserved, but the entropy of the entire system increases because of these interactions.
H=He+Hph+He−ph