quantum mechanics basic

1. Summer Physics 2016: Attachment1 for the Second Update
Roa, F. J. P.
The following would just be a refresher on basic quantum mechanics as applied for statistical
mechanics. In other attachments to follow, we will show the same results using path integrals.
Say we are given a state vector | 𝑥⟩ for the position state x and in the discrete basis of | 𝑛⟩, this
can be represented by
(1)
| 𝑥⟩ = ∑| 𝑛⟩⟨ 𝑛| 𝑥⟩
𝑛
Then these discrete bases are assumed to observe the completeness relation
(2)
1 = ∑| 𝑛⟩⟨ 𝑛|
𝑛
Let us operate this state vector with a time evolution operator exp(−𝛽𝐻̂ ) as expressed in the
analytic continuation, where 𝑇 = −𝑖 Τ = −ℏβ, and by some conversion factor we identify the
inverse temperature as β =
1
𝑘 𝐵 𝑇𝑒𝑚𝑝
(3)
exp(−𝛽𝐻̂ ) | 𝑥⟩ = ∑ ∑| 𝑛′⟩⟨𝑛′| exp(−𝛽𝐻̂ )|𝑛⟩⟨ 𝑛| 𝑥⟩
𝑛𝑛′
By Mclaurin expansion, we can write the time evolution operator as
(4)
exp(−𝛽𝐻̂ ) = 1 − 𝛽𝐻̂ + ∑
1
𝑞!
(−𝛽𝐻̂)
𝑞
∞
𝑞=2

2. It is assumed here that the harmonic oscillator is a bosonic oscillator whose Hamiltonian operator
𝐻̂ can be expressed in terms of the raising and lowering operators
(5)
𝐻̂ = ℏ𝜔 (𝑎†
𝑎 +
1
2
)
This operator satisfies the eigenvalue equation
(6)
𝐻̂ | 𝑛⟩ = ℏ𝜔 (𝑛 +
1
2
) | 𝑛⟩ = 𝐸 𝑛 | 𝑛⟩
We can raise this Hamiltonian operator to any given power q and let it operate on the state vector
| 𝑛⟩. Such gives
(7)
𝐻̂ 𝑞 | 𝑛⟩ = 𝐸 𝑛
𝑞
| 𝑛⟩
Given these results and applying them in the Mclaurin expansion (4) as we have operated then
this time evolution operator on | 𝑛⟩ we get the corresponding eigenvalue equation
(8)
exp(−𝛽𝐻̂ ) | 𝑛⟩ = exp(−𝛽𝐸 𝑛 )| 𝑛⟩
where exp(−𝛽𝐸 𝑛 ) are the eigenvalues of the operator exp(−𝛽𝐻̂ ) with state vector | 𝑛⟩.
Following this we obtain for the matrix elements needed in (3)
(9)
⟨𝑛′|exp(−𝛽𝐻̂ ) |𝑛⟩ = exp(−𝛽𝐸 𝑛 )⟨ 𝑛′| 𝑛⟩
in which we add the orthonormality condition of the discrete basis,
(10)
⟨ 𝑛′| 𝑛⟩ = 𝛿 𝑛′
𝑛
With these matrix elements (9) we would be able to write (3) as

3. (11)
exp(−𝛽𝐻̂ )| 𝑥⟩ = ∑| 𝑛′⟩exp(−𝛽𝐸 𝑛′ )⟨ 𝑛′| 𝑥⟩
𝑛′
thus, also obtaining the corresponding matrix elements in the basis of | 𝑥⟩ as expressed using the
projection of the result above on some arbitrary state vector | 𝑥′⟩.
(12)
⟨𝑥′
|exp(−𝛽𝐻̂ ) |𝑥⟩ = ∑⟨ 𝑥′| 𝑛′⟩ exp(−𝛽𝐸 𝑛′ )⟨ 𝑛′| 𝑥⟩
𝑛′
We can then obtain for the trace of these matrix elements by integration over the position
variable 𝑥′
= 𝑥. The integral limits are ofcourse, −∞ and ∞. This would give
(13)
∫ 𝑑𝑥 ⟨𝑥| exp(−𝛽𝐻̂ ) |𝑥⟩ = ∑∫ 𝑑𝑥 ⟨ 𝑛| 𝑥⟩⟨ 𝑥| 𝑛⟩exp(−𝛽𝐸 𝑛 ) = ∑exp(−𝛽𝐸 𝑛 )
𝑛𝑛
as we would also observe the completeness relation in the continuous basis | 𝑥⟩
(13.1)
1 = ∫| 𝑥⟩ 𝑑𝑥⟨ 𝑥|
giving the result
(13.2)
∫ 𝑑𝑥 ⟨ 𝑛| 𝑥⟩⟨ 𝑥| 𝑛⟩ = ⟨ 𝑛| 𝑛⟩ = 𝛿 𝑛𝑛 = 1
Trace (13) represents the partition function over the discrete energies 𝐸 𝑛.
In the next attachments, we will use path integration to show this same result.

4. Ref
[1]Merzbacher, E., Quantum Mechanics, 2nd edition, 1970, Wiley & Sons, Inc.
[2] Baal, P., A COURSE IN FIELD THEORY,
http://www.lorentz.leidenuniv.nl/~vanbaal/FTcourse.html