Quantum Physics CLASS Section V
The number representation, Creation and annihilation operators.
Zad. 1) Consider a state |ξ1(x1), ξ2(x2)i consisting of two non interacting particles where ξ1(x1), ξ2(x2) describe particular states; lower indexes denote the number of a particle and upper indexes correspond to quantum num- bers which determined a state. Define an operation which maps the state
|ξ1(x1), ξ2(x2)i into antisymmetric state, i.e. interchanging of two particle leads to the change of sign for the wave function.
Zad. 1) Creation and annhilation operators for fermions.
a)Let ˆnkdenote the number particle operator (for fermions) and |ν1, ν2, ..., νk, ...i, νi = 0, 1, be the particle occupation state. Furthermore, suppose that ˆak and ˆ
a+k (adjoint of ˆak) satisfy the relations
{ˆak, ˆal} = 0 , {ˆa+k, ˆa+l } = 0 , {ˆa+k, ˆal} = δkl. Show that ˆa+kˆak≡ ˆnk.
b) Show that
ak|0i = 0 , where |0i- vacuum state. Moreover prove
ˆ
ak|0 , , ..., 1k, ...i = |0i , ˆ
a+k|0i = |0 , , ..., 1k, ...i .
Coherent states.
3) Consider the bosonic creation and annihilation operators: ˆa+ and ˆa (i.e.
[ˆa, ˆa+] = 1) and n-bosonic (non-normalized) state determined by
|ni = (ˆa+)n|0i . 1
Show that
ˆ
a+|ni =√
n + 1|n + 1i and
ˆ
a|ni =√
n|n − 1i
(b) Suppose that |αi determines an eigenvector of ˆa to (complex) eigenvalue α, i.e.
ˆ
a|αi = α|αi . Show that
|αi = e−|α|2/2X
n
αn
√n!|ni .
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