Quantum Mechanics

Abstract

After recalling some basic concepts of statistical physics and quantum mechanics, the partition function of a harmonic oscillator is defined and evaluated in the standard canonical formalism. An imaginary-time path integral representation is subsequently developed for the partition function, the path integral is evaluated in momentum space, and the earlier result is reproduced upon a careful treatment of the zero-mode contribution. Finally, the concept of 2-point functions (propagators) is introduced, and some of their key properties are derived in imaginary time.

Appendix: 2-Point Function

Defining a Heisenberg-like operator (with it → τ)

$$\displaystyle{ \hat{x}(\tau )\; \equiv \; e^{\frac{\hat{H}\tau } {\hslash } }\hat{x}\,e^{-\frac{\hat{H}\tau } {\hslash } }\;,\quad 0 <\tau <\beta \hslash \;, }$$

(1.59)

we define a “2-point Green’s function” or a “propagator” through

$$\displaystyle{ G(\tau )\; \equiv \; \frac{1} {\mathcal{Z}}\,\mathrm{Tr}\,{\Bigl [e^{-\beta \hat{H}}\hat{x}(\tau )\hat{x}(0)\Bigr ]}\;. }$$

(1.60)

The corresponding path integral can be shown to read

$$\displaystyle{ G(\tau ) = \frac{\int _{x(\beta \hslash )=x(0)}\!\mathcal{D}x\,x(\tau )x(0)\exp [-S_{E}^{}/\hslash ]} {\int _{x(\beta \hslash )=x(0)}\!\mathcal{D}x\,\exp [-S_{E}^{}/\hslash ]} \;, }$$

(1.61)

whereby the normalization of $\mathcal{D}x$ plays no role. In the following, we compute G(τ) explicitly for the harmonic oscillator, by making use of

• (a) the canonical formalism, i.e. expressing $\hat{H}$ and $\hat{x}$ in terms of the annihilation and creation operators $\hat{a}$ and $\hat{a}^{\dag }$,

• (b) the path integral formalism, working in Fourier space.

Starting with the canonical formalism, we write all quantities in terms of $\hat{a}$ and $\hat{a}^{\dag }$:

$$\displaystyle{ \hat{H} = \hslash \omega \,{\Bigl (\hat{a}^{\dag }\hat{a} + \frac{1} {2}\Bigr )}\;,\quad \hat{x} = \sqrt{ \frac{\hslash } {2m\omega }}(\hat{a} +\hat{ a}^{\dag })\;,\quad [\hat{a},\hat{a}^{\dag }] = 1\;. }$$

(1.62)

In order to construct $\hat{x}(\tau )$, we make use of the expansion

$$\displaystyle{ e^{\hat{A}}\hat{B}\,e^{-\hat{A}} =\hat{ B} + [\hat{A},\hat{B}] + \frac{1} {2!}[\hat{A},[\hat{A},\hat{B}]] + \frac{1} {3!}[\hat{A},[\hat{A},[\hat{A},\hat{B}]]] +\ldots \;. }$$

(1.63)

Noting that

$$\displaystyle\begin{array}{rcl} [\hat{H},\hat{a}]& =& \hslash \omega [\hat{a}^{\dag }\hat{a},\hat{a}] = -\hslash \omega \hat{a}\;, \\{} [\hat{H},[\hat{H},\hat{a}]]& =& (-\hslash \omega )^{2}\hat{a}\;, \\{} [\hat{H},\hat{a}^{\dag }]& =& \hslash \omega [\hat{a}^{\dag }\hat{a},\hat{a}^{\dag }] = \hslash \omega \hat{a}^{\dag }\;, \\{} [\hat{H},[\hat{H},\hat{a}^{\dag }]]& =& (\hslash \omega )^{2}\hat{a}^{\dag }\;, {}\end{array}$$

(1.64)

and so forth, we can write

$$\displaystyle\begin{array}{rcl} e^{\frac{\hat{H}\tau } {\hslash } }\hat{x}\,e^{-\frac{\hat{H}\tau } {\hslash } }& =& \sqrt{ \frac{\hslash } {2m\omega }}\,\biggl \{\hat{a}{\biggl [1 -\omega \tau + \frac{1} {2!}(-\omega \tau )^{2} +\ldots \biggr ]} \\ & & +\hat{a}^{\dag }{\biggl [1 +\omega \tau + \frac{1} {2!}(\omega \tau )^{2} +\ldots \biggr ]}\biggr \} \\ & =& \sqrt{ \frac{\hslash } {2m\omega }}\,{\Bigl (\hat{a}\,e^{-\omega \tau } +\hat{ a}^{\dag }e^{\omega \tau }\Bigr )}\;. {}\end{array}$$

(1.65)

Inserting now $\mathcal{Z}$ from Eq. (1.17), Eq. (1.60) becomes

$$\displaystyle{ G(\tau ) = 2\sinh {\Bigl (\frac{\beta \hslash \omega } {2}\Bigr )}\sum _{n=0}^{\infty }\langle n\vert e^{-\beta \hslash \omega (n+\frac{1} {2} )} \frac{\hslash } {2m\omega }{\Bigl (\hat{a}\,e^{-\omega \tau } +\hat{ a}^{\dag }e^{\omega \tau }\Bigr )}{\bigl (\hat{a} +\hat{ a}^{\dag }\bigr )}\vert n\rangle \;. }$$

(1.66)

With the relations $\hat{a}^{\dag }\vert n\rangle = \sqrt{n + 1}\vert n + 1\rangle$ and $\hat{a}\vert n\rangle = \sqrt{n}\vert n - 1\rangle$ we can identify the non-zero matrix elements,

$$\displaystyle{ \langle n\vert \hat{a}\hat{a}^{\dag }\vert n\rangle = n + 1\;,\quad \langle n\vert \hat{a}^{\dag }\hat{a}\vert n\rangle = n\;. }$$

(1.67)

Thereby we obtain

$$\displaystyle{ G(\tau ) = \frac{\hslash } {m\omega }\sinh {\Bigl (\frac{\beta \hslash \omega } {2}\Bigr )}\exp {\Bigl ( -\frac{\beta \hslash \omega } {2}\Bigr )}\sum _{n=0}^{\infty }e^{-\beta \hslash \omega n}{\Bigl [e^{-\omega \tau } + n{\Bigl (e^{-\omega \tau } + e^{\omega \tau }\Bigr )}\Bigr ]}\;, }$$

(1.68)

where the sums are quickly evaluated as geometric sums,

$$\displaystyle\begin{array}{rcl} \sum _{n=0}^{\infty }e^{-\beta \hslash \omega n}& =& \frac{1} {1 - e^{-\beta \hslash \omega }} \;, \\ \sum _{n=0}^{\infty }ne^{-\beta \hslash \omega n}& =& -\frac{1} {\beta \hslash } \frac{\mathrm{d}} {\mathrm{d}\omega } \frac{1} {1 - e^{-\beta \hslash \omega }} = \frac{e^{-\beta \hslash \omega }} {(1 - e^{-\beta \hslash \omega })^{2}}\;.{}\end{array}$$

(1.69)

In total, we then have

$$\displaystyle\begin{array}{rcl} G(\tau )& =& \frac{\hslash } {2m\omega }{\Bigl (1 - e^{-\beta \hslash \omega }\Bigr )}{\biggl [ \frac{e^{-\omega \tau }} {1 - e^{-\beta \hslash \omega }} +{\Bigl ( e^{-\omega \tau } + e^{\omega \tau }\Bigr )} \frac{e^{-\beta \hslash \omega }} {(1 - e^{-\beta \hslash \omega })^{2}}\biggr ]} \\ & =& \frac{\hslash } {2m\omega } \frac{1} {1 - e^{-\beta \hslash \omega }} {\Bigl [e^{-\omega \tau } + e^{\omega (\tau -\beta \hslash )}\Bigr ]} \\ & =& \frac{\hslash } {2m\omega } \frac{\cosh \left [\left (\frac{\beta \hslash } {2} -\tau \right )\omega \right ]} {\sinh \left [\frac{\beta \hslash \omega } {2} \right ]} \;. {}\end{array}$$

(1.70)

As far as the path integral treatment goes, we employ the same representation as in Eq. (1.50), noting that C′ drops out in the ratio of Eq. (1.61). Recalling the Fourier representation of Eq. (1.45),

$$\displaystyle\begin{array}{rcl} x(\tau )& =& T{\biggl \{a_{0} +\sum _{ k=1}^{\infty }{\biggl [(a_{ k} + ib_{k})e^{i\omega _{k}\tau } + (a_{k} - ib_{k})e^{-i\omega _{k}\tau }\biggr ]}\biggr \}}\;,{}\end{array}$$

(1.71)

$$\displaystyle\begin{array}{rcl} x(0)& =& T{\biggl \{a_{0} +\sum _{ l=1}^{\infty }2a_{ l}\biggr \}}\;,{}\end{array}$$

(1.72)

the observable of our interest becomes

$$\displaystyle{ G(\tau )\; =\;{\bigl \langle\, x(\tau )x(0)\,\bigr \rangle}\; \equiv \; \frac{\int \!\mathrm{d}a_{0}\int \!\prod _{n\geq 1}\mathrm{d}a_{n}\,\mathrm{d}b_{n}\,x(\tau )\,x(0)\exp [-S_{E}^{}/\hslash ]} {\int \!\mathrm{d}a_{0}\int \!\prod _{n\geq 1}\mathrm{d}a_{n}\,\mathrm{d}b_{n}\,\exp [-S_{E}^{}/\hslash ]} \;. }$$

(1.73)

At this point, we employ the fact that the exponential is quadratic in $a_{0},a_{n},b_{n} \in \mathbb{R}$, which immediately implies

$$\displaystyle{ \langle a_{0}a_{k}\rangle =\langle a_{0}b_{k}\rangle =\langle a_{k}b_{l}\rangle = 0\;,\quad \langle a_{k}a_{l}\rangle =\langle b_{k}b_{l}\rangle \propto \delta _{kl}\;, }$$

(1.74)

with the expectation values defined in the sense of Eq. (1.73). Thereby we obtain

$$\displaystyle{ G(\tau ) = T^{2}\Bigl \langle a_{ 0}^{2} +\sum _{ k=1}^{\infty }2a_{ k}^{2}\left (e^{i\omega _{k}\tau } + e^{-i\omega _{k}\tau }\right )\Bigr \rangle\;, }$$

(1.75)

where

$$\displaystyle\begin{array}{rcl} \langle a_{0}^{2}\rangle & =& \frac{\int \!\mathrm{d}a_{0}\,a_{0}^{2}\,\exp \left (-\frac{1} {2}mT\omega ^{2}a_{ 0}^{2}\right )} {\int \!\mathrm{d}a_{0}\,\exp \left (-\frac{1} {2}mT\omega ^{2}a_{0}^{2}\right )} \\ & =& - \frac{2} {m\omega ^{2}} \frac{\mathrm{d}} {\mathrm{d}T}{\biggl [\ln \int \!\mathrm{d}a_{0}\,\exp \left (-\frac{1} {2}mT\omega ^{2}a_{ 0}^{2}\right )\biggr ]} = - \frac{2} {m\omega ^{2}} \frac{\mathrm{d}} {\mathrm{d}T}{\biggl [\ln \sqrt{ \frac{2\pi } {m\omega ^{2}T}}\,\biggr ]} \\ & =& \frac{1} {m\omega ^{2}T}\;, {}\end{array}$$

(1.76)

$$\displaystyle\begin{array}{rcl} \langle a_{k}^{2}\rangle & =& \frac{\int \!\mathrm{d}a_{k}\,a_{k}^{2}\,\exp \left [-mT(\omega _{ k}^{2} +\omega ^{2})a_{ k}^{2}\right ]} {\int \!\mathrm{d}a_{k}\,\exp \left [-mT(\omega _{k}^{2} +\omega ^{2})a_{k}^{2}\right ]} \\ & =& \frac{1} {2m(\omega _{k}^{2} +\omega ^{2})T}\;. {}\end{array}$$

(1.77)

Inserting these into Eq. (1.75) we get

$$\displaystyle{ G(\tau ) = \frac{T} {m}{\biggl (\frac{1} {\omega ^{2}} +\sum _{ k=1}^{\infty }\frac{e^{i\omega _{k}\tau } + e^{-i\omega _{k}\tau }} {\omega _{k}^{2} +\omega ^{2}} \biggr )} = \frac{T} {m}\sum _{k=-\infty }^{\infty } \frac{e^{i\omega _{k}\tau }} {\omega _{k}^{2} +\omega ^{2}}\;, }$$

(1.78)

where we recall that $\omega _{k} = 2\pi kT/\hslash$.

There are various ways to evaluate the sum in Eq. (1.78). We encounter a generic method in Sect. 2.2, so let us present a different approach here. We start by noting that

$$\displaystyle{ {\biggl (-\frac{\mathrm{d}^{2}} {\mathrm{d}\tau ^{2}} +\omega ^{2}\biggr )}G(\tau ) = \frac{T} {m}\sum _{k=-\infty }^{\infty }e^{i\omega _{k}\tau } = \frac{\hslash } {m}\,\delta (\tau \mathop{\mbox{ mod}}\beta \hslash )\;, }$$

(1.79)

where we made use of the standard summation formula $\sum _{k=-\infty }^{\infty }e^{i\omega _{k}\tau } =\beta \hslash \,\delta (\tau \mathop{\mbox{ mod}}\beta \hslash )$.¹

Next, we solve Eq. (1.79) for $0 <\tau <\beta \hslash$, obtaining

$$\displaystyle{ {\biggl (-\frac{\mathrm{d}^{2}} {\mathrm{d}\tau ^{2}} +\omega ^{2}\biggr )}G(\tau ) = 0\quad \Rightarrow \quad G(\tau ) = A\,e^{\omega \tau } + B\,e^{-\omega \tau }\;, }$$

(1.80)

where A, B are unknown constants. The solution can be further restricted by noting that the definition of G(τ), Eq. (1.78), indicates that $G(\beta \hslash -\tau ) = G(\tau )$. Using this condition to obtain B, we then get

$$\displaystyle{ G(\tau ) = A\left [e^{\omega \tau } + e^{\omega (\beta \hslash -\tau )}\right ]\;. }$$

(1.81)

The remaining unknown A can be obtained by integrating Eq. (1.79) over the source at τ = 0 and making use of the periodicity of G(τ), $G(\tau +\beta \hslash ) = G(\tau )$. This finally produces

$$\displaystyle{ G'((\beta \hslash )^{-}) - G'(0^{+}) = \frac{\hslash } {m}\quad \Rightarrow \quad 2\omega A\left (e^{\omega \beta \hslash } - 1\right ) = \frac{\hslash } {m}\;, }$$

(1.82)

which together with Eq. (1.81) yields our earlier result, Eq. (1.70).

The agreement of the two different computations, Eqs. (1.60) and (1.61), once again demonstrates the equivalence of the canonical and path integral approaches to solving thermodynamic quantities in a quantum-mechanical setting.