10.4: A simple example - the quantum harmonic oscillator
- Page ID
- 5259
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)As a simple example of the trace procedure, let us consider the quantum harmonic oscillator. The Hamiltonian is given by
\[ H = {P^2 \over 2m} + {1 \over 2}m\omega^2 X^2 \nonumber \]
and the eigenvalues of \(H\) are
\[ E_n = \left(n + {1 \over 2}\right)\hbar\omega,\;\;\;\;\;\;\;\;\;\;n=0,1,2,... \nonumber \]
Thus, the canonical partition function is
\[Q(\beta) = \sum_{n=0}^{\infty} e^{-\beta (n+1/2)\hbar\omega} = e^{-\beta \hbar \omega/2}\sum_{n=0}^{\infty}\left(e^{-\beta\hbar\omega}\right)^n \nonumber \]
This is a geometric series, which can be summed analytically, giving
\[ Q(\beta) = {e^{-\beta \hbar\omega/2} \over 1-e^{-\beta \hbar \omega}} = {1 \over e^{\beta \hbar \omega / 2} - e^{-\beta \hbar\omega/2}}= {1 \over 2}csch(\beta\hbar\omega/2) \nonumber \]
The thermodynamics derived from it as as follows:
- 1.
- Free energy:
The free energy is
\[A = -{1 \over \beta}\ln Q(\beta) = {\hbar\omega \over 2} +{1 \over \beta}\ln \left(1-e^{-\beta \hbar \omega}\right) \nonumber \]
- 2.
- Average energy:
The average energy \(E = \langle H \rangle \) is
\[ E = -{\partial \over \partial \beta}\ln Q(\beta)= {\hbar\omega \over 2} + {\hbar \omega e^{-\beta \hbar \omega}}= \left({1 \over 2} + \langle n \rangle \right)\hbar\omega \nonumber \]
- 3.
- Entropy
The entropy is given by
\[ S = k\ln Q(\beta) + {E \over T} =-k\ln \left(1-e^{-\beta \hbar \omega} \right ) + {\hbar \omega \over T}{e^{-\beta \hbar \omega}\over 1-e^{-\beta \hbar \omega}} \nonumber \]
Now consider the classical expressions. Recall that the partition function is given by
\[ Q(\beta) = {1 \over h} \int dp dx e^{-\beta \left({p^2 \over 2m} + {1 \over 2} m\omega ^2 x^2 \right )} = {1 \over h} \left ( {2 \pi m \over \beta} \right )^{1/2} = {2\pi \over \beta \omega h} ={1 \over \beta \hbar \omega} \nonumber \]
Thus, the classical free energy is
\[A_{\rm cl} = {1 \over \beta}\ln(\beta \hbar \omega) \nonumber \]
In the classical limit, we may take \(\hbar \) to be small. Thus, the quantum expression for \(A\) becomes, approximately, in this limit:
\[A_{\rm Q} \longrightarrow {\hbar \omega \over 2} + {1 \over \beta}\ln (\beta \hbar \omega) \nonumber \]
and we see that
\[A_{\rm Q} - A_{\rm cl} \longrightarrow {\hbar \omega \over 2} \nonumber \]
The residual \({\hbar \omega \over 2} \) (which truly vanishes when \(\hbar \rightarrow 0 \)) is known as the quantum zero point energy. It is a pure quantum effect and is present because the lowest energy quantum mechanically is not \(E = 0 \) but the ground state energy \(E={\hbar\omega \over 2} \).