Chapter 1

REVIEW of QUANTUM MECHANICS

Key Concepts:

Hilbert space of states; Observables; Evaluation operator; Along a contour; Imaginary time; Chronological Order; Schrödinger-Heisenberg interaction; Picture representation; Schröedinger picture, states evolve in time according:

i∂|ψ_{s}(t) >\overline{∂t} = H_{s}(t)|ψ_{s}(t) >

|ψ_{s}(t) >= U_{s}(t, t_{0})|ψ_{s}(t_{0}) >

U_{s}(t, t_{0}) evelution operator plays an essential role in the theroy.

i∂U(t, t_{0} \overline{∂t} = H_{s}(t)U(t, t_{0})

U(t, t_{0}) = I

For mixed states, use density matrix:

idρ_{s} \overline{dt} = [H_{s}, ρ_{s}]

Figure 1.1:

Formal solution of the evolution equation:

U(t, t_{0}) = T e^{\frac{1}{i} ∫_{t_{0}}^{t} H_{s}(τ)dτ} = ∑_{n=0}^{∞} \frac{(1}{i)^{n}} ∫_{t_{0}}^{t} … ∫_{t_{0}}^{t_{n−1}} H_{s}(t_{1}) … H_{s}(t_{n})dt_{1} … dt_{n} The operator orders operatorrs putting the latter ones on the right. Eqn.( )–( ) can be in fact formulated for an arbituary oriented contour

C in the condex plane if we assume that H_{s} is defined there.

If C is a contour going from Z_{1} to Z_{2} :

U(Z_{2}, Z_{1}) ≡ T_{C}e^{\frac{1}{i} ∫_{C} H_{s}(z) dz}

obeys

i∂U(t_{2}, t_{1})\overline{∂Z_{2}} = H_{s}(Z_{2})U(Z_{2}, Z_{1})

T_{C} orders operators along the contour C putting latter operators on the right.

Heisenberg picture. Here one performs a unitary rotation to keep the state vectors fixed,i.e.

Figure 1.2: figure 2

|ψ_{H}(t) > = U(t, t_{0})^{−1}|ψ_{s}(t) >

t_{0} is an arbituary time where all the pictures ? made. The observables evolve according to O_{H}(t) = U(t, t_{0})^{−1}O_{s}(t)U(t, t_{0})

Ȯ_{H} = ∂O_{H} \overline{∂t} + \frac{1}{i}[O_{H}, H_{H}]

where ∂O_{H} _{\overline{∂t}} ≡ U(t, t_{0})_{−1}∂O_{S} _{\overline{∂t}}U(t, t_{0}) The density matrix obeys dρ_{H} _{\overline{dt}} = 0 The notion of evolution along a contour is useful because it allows to

unify different formulations of many body methods. A particular useful contour is [0, -iβ], [i.e.]alongtheimagingaxis.(seeF ig.2)

Example: The evolution operator from 0 to −iβalong the contour shown in Fig. 2 is given by:

U_{S}(−iβ, 0) = T_{C}e^{\frac{1}{i} ∫_{C} H_{S}(τ) dτ′}

and equals to the Boltzmann density matrix e^{−βH_{S}} if H_{S} does not depend on time.

Interaction picture: This representation is defined with reference to a ?thing of the hamito-

nian into two parts H = H_{0} + V. Roughly speaking, the onservables and evolved which is usually simple while the state vectors are driven by V.

|ψ_{I}(t) >= e^{iH_{0}(t_{1}−t_{0})}|ψ_{S}(t) >

The observables evolve according to V_{I}(t) = e^{iH_{0}(t−t_{0})}V_{S}(t)^{−iH_{0}(t−t_{0})}. The evolution operator in this picture is given by:

U_{I}(t_{i}, t) = e^{iH_{0}(t_{1}−t_{0})}U_{S}(t_{1}, t)e^{−iH_{0}(t−t_{0})}

and obeys:

i∂U_{I}(t_{1}, t_{2})\overline{∂t} = V_{I}(t_{1})U_{I}(t_{1}, t_{2})
U(t, t) = I

The formal solution:

U_{I}(t_{1}, t_{2}) = T e^{\frac{1}{i} ∫_{t_{2}}^{t_{1}} V_{I}(τ) dτ}

similarly:

iȮ_{I} = [O_{I}, H_{0}]
i