91影视

The Boltzmann distribution (also called the Gibbs distribution) is a probability distribution or probability measure that gives the probability that a system will be in a certain state as a function of the energy of that state and the temperature of the system.

In quantum mechanics, the standard deviation is nothing but the uncertainty in position

$\mathrm{螖虫}鈮�\sqrt{鉄�{\mathrm{x}}^2鉄�鈭�{鉄�\mathrm{x}鉄�}^2}$

Where, $鉄�{\mathrm{x}}^2鉄�$is the expectation value of the square of position, and $鉄�\mathrm{x}鉄�$ is the expectation value of the position.

The expectation value of the square of position is

$鉄�{\mathrm{x}}^2鉄�={鈭�}_{鈭�\mathrm{鈭�}}^{\mathrm{鈭�}}{\mathrm{x}}^2{\left|\mathrm{蠄}\left(\mathrm{x}\right)\right|}^2\mathrm{dx}$

In infinite potential well, the general solution of Schrodinger鈥檚 time-independent equation is

$\mathrm{蠄}\left(\mathrm{x}\right)=\sqrt{\frac{2}{\mathrm{L}}}\sin \left(\frac{\mathrm{苍蟺虫}}{\mathrm{L}}\right)$

Inserting this in the expectation value expression

$\begin{array}{c}鉄�{\mathrm{x}}^2鉄�={鈭�}_{鈭�\mathrm{鈭�}}^{\mathrm{鈭�}}{\mathrm{x}}^2{\left(\sqrt{\frac{2}{\mathrm{L}}}\sin \left(\frac{\mathrm{苍蟺虫}}{\mathrm{L}}\right)\right)}^2\mathrm{dx}\\ =\frac{2}{\mathrm{L}}{鈭�}_0^{\mathrm{L}}{\mathrm{x}}^2{\sin }^2\left(\frac{\mathrm{苍蟺虫}}{\mathrm{L}}\right)\mathrm{dx}\text{鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌�}\left(\mathrm{蠄}\left(\mathrm{x}\right)\text{鈥�}\mathrm{is}\text{鈥�}\mathrm{zero}\text{鈥�}\mathrm{for}\text{鈥�}\mathrm{any}\text{鈥�}\mathrm{region}\text{鈥�}\mathrm{except}\text{鈥�}[0,\mathrm{L}]\right)\\ =\frac{2}{\mathrm{L}}{鈭�}_0^{\mathrm{L}}\frac{{\mathrm{x}}^2}{2}\left(1鈭�\cos \left(\frac{2\mathrm{苍蟺虫}}{\mathrm{L}}\right)\right)\end{array}$

$\begin{array}{c}鉄�{\mathrm{x}}^2鉄�=\frac{1}{\mathrm{L}}{鈭�}_0^{\mathrm{L}}{\mathrm{x}}^2鈭�\frac{1}{\mathrm{L}}{鈭�}_0^{\mathrm{L}}{\mathrm{x}}^2\cos \left(\frac{2\mathrm{苍蟺虫}}{\mathrm{L}}\right)\\ =\frac{{\mathrm{L}}^2}{3}鈭�\frac{{\mathrm{L}}^2}{2{\mathrm{n}}^2{\mathrm{蟺}}^2}\text{鈥夆赌夆赌夆赌夆赌�}\left[\mathrm{Use}\text{鈥�}\mathrm{Integration}\text{鈥�}\mathrm{By}\text{鈥�}\mathrm{parts}\text{鈥�}\mathrm{for}\text{鈥�}\mathrm{second}\text{鈥�}\mathrm{integral}\right]\end{array}$

The expectation value is,

$\begin{array}{c}鉄�x鉄�={鈭�}_{鈭�\mathrm{鈭�}}^{\mathrm{鈭�}}x{\left|蠄\left(x\right)\right|}^{2}dx\\ =\frac{2}{L}{鈭�}_0^L{x}^2{\sin }^2\left(\frac{n蟺x}{L}\right)dx\text{鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌�}\left(蠄\left(x\right)\text{鈥�}is\text{鈥�}zero\text{鈥�}for\text{鈥�}any\text{鈥�}region\text{鈥�}except\text{鈥�}[0,L]\right)\\ =\frac{2}{L}{鈭�}_0^L\frac{x}{2}\left[1鈭�\cos \left(\frac{2蟺nx}{L}\right)\right]\\ =\frac{1}{L}{鈭�}_0^{L}xdx鈭�\frac{1}{L}{鈭�}_0^{L}x\cos \left(\frac{2蟺nx}{L}\right)dx\end{array}$

$鉄�\mathrm{x}鉄�=\frac{\mathrm{L}}{2}鈭�0\text{鈥夆赌夆赌夆赌夆赌�}\left[\mathrm{Use}\text{鈥�}\mathrm{Integration}\text{鈥�}\mathrm{By}\text{鈥�}\mathrm{parts}\text{鈥�}\mathrm{for}\text{鈥�}\mathrm{second}\text{鈥�}\mathrm{integral}\right]$

Therefore, the uncertainty in the measurement of position is,

$\begin{array}{c}\mathrm{螖虫}=\sqrt{\frac{{\mathrm{L}}^3}{3}鈭�\frac{{\mathrm{L}}^2}{2{\mathrm{n}}^2{\mathrm{蟺}}^2}鈭�\frac{{\mathrm{L}}^2}{4}}\\ =\sqrt{\frac{{\mathrm{L}}^2}{12}鈭�\frac{{\mathrm{L}}^2}{2{\mathrm{n}}^2{\mathrm{蟺}}^2}}\\ =\mathrm{L}\sqrt{\frac{1}{12}鈭�\frac{1}{2{\mathrm{n}}^2{\mathrm{蟺}}^2}}\end{array}$

The general uncertainty term depends on the principle quantum number $n$. Higher the value of $n$higher will be the uncertainty in the measurement of the position.

For a very high value of the quantum number, the second term in the root can be neglected, and therefore you will get,

$\mathrm{螖虫}=\frac{\mathrm{L}}{\sqrt{12}}$

This is true for classical limits as shown in Exercise $55$.