凝聚态和原子物理中的多体现象
8.514: Many-body phenomena in condend matter and atomic physics Problem Set #6 Due: 10/21/03
8.514: 凝聚态和原子物理中的多体现象 问题集 # 6提交日期：10/21/03
Bardeen – Cooper - Schrieer theory
巴丁-库柏-施里弗理论
1. Quasiparticles.
准粒子
Consider quasiparticles of a BCS superconductor,
考虑BCS超导体准粒子
where and , etc., are Bogoliubov quasiparticle operators.
其中， 和等是巴格寥夫准粒子算符。
a) Find out how many particles are contained in one quasiparticle. For that, consider a state with one quasiparticle added to the BCS ground state.
找出一个准粒子中包含的粒子数。为此，考虑将一个准粒子加到BCS基态时的态。
evaluate the expectation value
估算期望值
and express the result in terms of the Bogoliubov angleθp. Can〈N〉be nagative? Explain.
并将结果用巴格寥夫角θp来表述。〈N〉可以取负值？解析该结果。
b) Consider momentum and spin of a quasiparticle in the state (2). What are they? Do they depend on the Bogoliubov angle?
考虑态(2)中的准粒子动量及自旋。它们是什么？它们依赖于巴格寥夫角吗？
2.Gap equation.
能隙方程
For a BCS superconductor derive the gap equation
推导BCS超导的间隔方程
with E* the interaction cutoff parameter (E* ~ EF for nonretarded contact interaction).
其中E* 是作用切断参数（E* ~ EF 非延迟接触作用）
Study the gap Δ as a function of temperature. Show that Δ decreas monotonically with T and vanishes at a certain temperature T = Tc. Find the value Tc.
研究关于温度的函数Δ。证明Δ随着T单调减小同时在T = Tc时为零。找出Tc值
3. Gap suppression in a superflow. Critical current.
在超流中的能隙压缩。临界流
Superflow in a superconductor is described by the order parameter with spatially varying pha, Δ(r)= Δ e2iqr , which is related to the superflow velocity by vs = q/m.
起导体中的超流是用随相随空间变化的序参数Δ(r)= Δ e2iqr来描述的，它与超流速通过vs = q/m 联系起来
BCS quasiparticles in the prence of superflow are described by the Hamiltonian
存在超流的情况BCS 准粒子由以下哈密顿量描述
which can be dioganalized by a Bogoliubov transformation in which the states p + q and –p+ q are paired up.
它可以通过态 p + q 和 –p+ q 成对的巴格寥夫变换来对角化。
a) Find the quasiparticle spectrum. Assuming |q|<< pF , show that the result can be interpreted in terms of Doppler shift E’p= Ep + vsp, where Ep is the spectrum in the abnce of the flow.
找出准粒子谱。假设|q|<< pF，证明结果可以用多谱勒频移E’p= Ep + vsp来解释，其中Ep 没有流动时的谱。
b) Show that the energy gap between the BCS ground state and the first excited state is reduced in the prence of the flow. Find the critical velocity at which the g
ap vanishes.
b)证明BCS基态及第一激发态之间的能隙在存在流动的情况下将被缩小。找出能隙消失的临界速度。
c) Consider BCS pairing in the frame co-moving with the flow. By using Galilean invariance, or otherwi, argue that the gap equation and thus the order parameter Δ are not affected by the flow. Combined with the result of part b) this shows that the energy gap and pairing amplitude Δ aint necessarily have to be equal. They happen to be equal in a clean superconductor in the abnce of external pair-breaking fields or flows, but are not equal in general.
c)考虑与流动同时移动的参考系下的BCS配对。利用伽利略不变性，或相反，说明能隙方程及由此所得的序参数Δ 不受流的影响。结合b)的结果，证明能隙及配对幅Δ不需要一定相等。它们只有在没有破坏配对的外场或流的干净超导体中才偶然相等，一般而言它们不相等。
