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
§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

§2.5 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
§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
§2.1 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
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
§2.1 Second quantization
§2.1 Second quantization

For Fermions
§2.1 Second quantization
§2.1 Second quantization
§2.1 Second quantization
k
(1 2 n

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

Second quantization
§2.1 Second quantization
§2.1 Second quantization
§2.1 Second quantization
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
§2.4 Landau phase transition theory
§2.4 Landau phase transition theory
§2.4 Landau phase transition theory
min, stable
max,instalble
phase transition point
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

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
§2.3 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
§2.5 Superfluidity theory
§2.5 Superfluidity theory

Bogoliubov approximate secondquantization method
§2.5 Superfluidity theory
§2.5 Superfluidity theory
§2.4 Landau phase transition theory
A~0 real, stable
img,forbidden
§2.4 Landau phase transition theory

Landau theory Ehrenfest equation
§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
§2.2 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
§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
§2.3 Superconductive theory
§2.3 Superconductive theory
§2.3 Superconductive theory
§2.3 Superconductive theory
§2.3 Superconductive theory

Energy gap equation
§2.5 Superfluidity theory
§2.5 Superfluidity theory
§2.5 Superfluidity theory
§2.5 Superfluidity theory
§2.5 Superfluidity theory
§2.5 Superfluidity theory
§2.3 Superconductive theory
§2.3 Superconductive theory
§2.3 Superconductive theory