苏汝铿高等量子力学讲义

§2.4 Landau phase transition theory
Van Laue criticism Can 2nd order phase transition exist?

§2.4 Landau phase transition theory
§2.4 Landau phase transition theory
§2.1 Second quantization
§2.1 Second quantization
§2.1 Second quantization
§2.1 Second quantization
Discussions The wave function is already symmetric nk is the particle number operator of k state
§2.5 Superfluidity theory
§2.5 Superfluidity theory
§2.5 Superfluidity theory
§2.5 Superfluidity theory
§2.5 Superfluidity theory
§2.5 Superfluidity theory
§2.1 Second quantization

We need to introduce the creation and the annihilation operators to deal with various problem in the many-body system
§2.1 Second quantization
§2.5 Superfluidity theory
§2.5 Superfluidity theory
§2.5 Superfluidity theory

§2.5 Superfluidity theory
Experiments: Superfluidity 10^-5~10^-4 cm (η0) κ∞ Mendelson effect λ- point

§2.5 Superfluidity theory
§2.5 Superfluidity theory
§2.2 Hartree-Fork mean field approximation
§2.2 Hartree-Fork mean field approximation
§2.3 Superconductive theory
Experimental results 1908 1911 K.Onnes Liquid helium Hg: Tc=4.2K
n!
i n
N!
C ( n1 , n 2 , ..., n k , ..., t )
§2.1 Second quantization
§2.1 Second quantization
§2.1 Second quantization
§2.1 Second quantization
§2.1 Second quantization
§2.4 Landau phase transition theory
§2.4 Landau phase transition theory
Landau theory Introducing “order parameter ”

p , T ,
§2.4 Landau phase transition theory
1933
1986
Meissner effect B=0
Muller, Bednorz High Tc Heat capacity Cs ~ exp(-Δ/kBT) Δ ~ energy gap Isotropic effect Tc M^1/2=const.
§2.3 Superconductive theory
§2.1 Second quantization
§2.1 Second quantization

For Fermions
§2.1 Second quantization
§2.1 Second quantization
§2.1 Second quantization
k
(1 2 n
§2.2 Hartree-Fork mean field approximation
§2.2 Hartree-Fork mean field approximation
§2.2 Hartree-Fork mean field approximation
§2.2 Hartree-Fork mean field approximation

§2.1 Second quantization
§2.1 Second quantization
§2.1 Second quantization

Second quantization
§2.1 Second quantization
§2.1 Second quantization
§2.1 Second quantization

Bose system
§2.1 Second quantization
n1 ,..., n k ,... ( r1 , ...rN )
ni ! N!

P
P k1 ( r1 )... k N ( rN )
§2.1 Second quantization
A ( k 1 , k 2 , ..., k n , t )
§2.3 Superconductive theory
§2.3 Superconductive theory
k k0
§2.3 Superconductive theory
§2.3 Superconductive theory
E0 E0
(N )
Stable state
§2.4 Landau phase transition theory

Ehrenfest equation
§2.4 Landau phase transition theory
§2.4 Landau phase transition theory
§2.4 Landau phase transition theory
§2.4 Landau phase transition theory
§2.4 Landau phase transition theory
§2.4 Landau phase transition theory
§2.4 Landau phase transition theory
§2.5 Superfluidity theory
Landau superfluidity theory New idea: elementary excitation
§2.2 Hartree-Fork mean field approximation
§2.2 Hartree-Fork mean field approximation
§2.2 Hartree-Fork mean field approximation

Spin effect
§2.2 Hartree-Fork mean field approximation
Uk 1
2
e
0
ik r
e
e dr r
2
2

0 0


2
e
0
ikr cos
1 r
r dr sin d d
2
4 e k
sin krdr
Screening Coulomb potential
Positive charge background cancels k=0 part
§2.3 Superconductive theory
§2.3 Superconductive theory
§2.3 Superconductive theory
§2.3 Superconductive theory

Bogoliubov-Valatin canonical transformation

Landau theory
§2.5 Superfluidity theory
§2.5 Superfluidity theory
§2.5 Superfluidity theory
§2.5 Superfluidity theory
§2.5 Superfluidity theory
§2.5 Superfluidity theory
mk
m
)
§2.2 Hartree-Fork mean field approximation
Key: two-body problem “one-body problem” + “mean field” Example: Free electron gas in the metal

§2.2 Hartree-Fork mean field approximation
§2.5 Superfluidity theory
§2.5 Superfluidity theory

Bogoliubov approximate secondquantization method
§2.5 Superfluidity theory
§2.5 Superfluidity theory
§2.3 Superconductive theory
§2.3 Superconductive theory
§2.3 Superconductive theory
§2.3 Superconductive theory
§2.3 Superconductive theory

Energy gap equation
§2.4 Landau phase transition theory
A~0 real, stable
img,forbidden
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§2.4 Landau phase transition theory

Landau theory Ehrenfest equation
§2.4 Landau phase transition theory
§2.4 Landau phase transition theory
§2.4 Landau phase transition theory
§2.4 Landau phase transition theory
min, stable
max,instalble
phase transition point
Chapter 2 Many Body Problem
§2.1 Second quantization

The identical particles cannot be distinguished
§2.1 Second quantization

The essence of the identical principle is that the state of a system should be described in terms of the particle number in a certain quantum state and the many-body problem should be discussed in the particle number representation instead of the original coordinate representation

Frohlisch Hamiltonian: e-p-e interaction Cooper pair: a k a k 0 BCS Theory (Variational method)
e-e attraction
Variational wave function
§2.3 Superconductive theory