91影视

The relativistic correction to the energy levels is:

${\mathit{E}}_{\mathbf{r}}^{\mathbf{1}}\mathbf{=}\frac{\mathbf{-}\mathbf{1}}{\mathbf{2}\mathbf{m}{\mathbf{c}}^{\mathbf{2}}}\left[E_n^2-2E_n鉄�\hat{V}鉄�+鉄�{\hat{V}}^2鉄�\right]$

For the harmonic oscillator

$$\begin{gathered} \hat{\mathbf{V}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathit{m}{\mathbf{蝇}}^{\mathbf{}\mathbf{2}\mathbf{}}\widehat{{\mathbf{x}}^{\mathbf{2}}\mathbf{}} \\ {\mathit{E}}_{\mathbf{n}}\mathbf{=}\mathbf{鈩�}\mathit{蝇}\left(n+\frac{1}{2}\right) \end{gathered}$$

Now, we can write the relativistic correction to the energy as:

$E_r^1=\frac{-1}{2m{c}^2}\left[{鈩�}^2{蝇}^2{\left(n+\frac{1}{2}\right)}^2-2鈩�蝇\left(n+\frac{1}{2}\right)鈰�\frac{1}{2}鈩�蝇\left(n+\frac{1}{2}\right)+\frac{1}{4}{m}^2{蝇}^4鉄�{\hat{x}}^4鉄�\right]$

From the example (2.5), we get:

$$\begin{gathered} {\hat{x}}^2=\frac{鈩�}{2m蝇}\left[a^{+}{a}^{+}+a^{+}a+a{a}^{+}+aa\right] \\ 鉄�\hat{V}鉄�=\frac{1}{2}鈩�蝇\left(n+\frac{1}{2}\right) \end{gathered}$$

Then

$E_r^1=\frac{-1}{2m{c}^2}\left[{鈩�}^2{蝇}^2{\left(n+\frac{1}{2}\right)}^2-{鈩�}^2{蝇}^2{\left(n+\frac{1}{2}\right)}^2+\frac{1}{4}{m}^2{蝇}^4鉄�{\hat{x}}^4鉄�\right]$

We need to find $鉄�{\hat{x}}^4鉄�$by calculating that:

$$\begin{gathered} {\hat{x}}^4=\frac{{鈩�}^2}{4m^2{蝇}^2}\left[(a^{+}{a}^{+}+a^{+}a+a{a}^{+}+aa)鈰�(a^{+}{a}^{+}+a^{+}a+a{a}^{+}+aa)\right] \\ 鉄�{\hat{x}}^4鉄�=鉄�n\left|x{^}^4\right|n鉄� \\ =\frac{{鈩�}^2}{4m^2{蝇}^2}[鉄�n|a^{+}{a}^{+}{a}^{+}{a}^{+}|n鉄�+鉄�n|a^{+}{a}^{+}{a}^{+}a|n鉄�+鉄�n|a^{+}{a}^{+}a{a}^{+}|n鉄�+鉄�n|a^{+}{a}^{+}aa|n鉄� \\ +鉄�n|a^{+}a{a}^{+}{a}^{+}|n鉄�+鉄�n|a^{+}a{a}^{+}a|n鉄� \\ +鉄�n|a^{+}aa{a}^{+}|n鉄�+鉄�n|a^{+}aaa|n鉄�+鉄�n|a{a}^{+}{a}^{+}{a}^{+}|n鉄�n \\ +鉄�n|a{a}^{+}{a}^{+}a|n鉄�+鉄�n|a{a}^{+}a{a}^{+}|n鉄�+鉄�n|a{a}^{+}aa|n鉄�+鉄�n|aa{a}^{+}{a}^{+}|n鉄�+鉄�n|aa{a}^{+}a|n鉄� \\ +鉄�n|aaa{a}^{+}|n鉄�+鉄�n|aaaa|n鉄� \end{gathered}$$

We know that:

$$\begin{gathered} a^{+}|n鉄�=\sqrt{n+1}|n+1鉄�a|n鉄� \\ =\sqrt{n}|n-1鉄� \end{gathered}$$

Where, the terms which contain equal number particles due to the raising and lowering operators can survive and the other terms equal to zero from${未}_{nm}=0鈥�;鈥�n鈮�m$

Then, the only terms which can be calculated are:

$$\begin{gathered} 鉄�n\left|a^{+}{a}^{+}aa\right|n鉄�=\sqrt{n}鉄�n\left|a^{+}{a}^{+}a\right|n-1鉄� \\ =\sqrt{n}\sqrt{(n-1)}鉄�n\left|a^{+}{a}^{+}\right|n-2鉄� \\ =\sqrt{n}(n-1)鉄�n\left|a^{+}\right|n-1鉄� \\ =n(n-1)鉄�n鈭�n鉄� \\ =n(n-1){未}_{nn} \\ =n(n-1) \end{gathered}$$

Solve further

$$\begin{gathered} 鉄�n\left|a^{+}a{a}^{+}a\right|n鉄�=鈭�n鉄�n\left|a^{+}a{a}^{+}\right|n-1鉄� \\ =n鉄�n\left|a^{+}a\right|n鉄� \\ =n鈭�n鉄�n\left|a^{+}\right|n-1鉄� \\ =n^2鉄�n鈭�n鉄� \\ =n^2{未}_{n}n \\ =n^2 \\ 鉄�n\left|a^{+}aa{a}^{+}\right|n鉄�=鈭�(n+1)鉄�n\left|a^{+}aa\right|n+1鉄� \\ =(n+1)鉄�n\left|a^{+}a\right|n鉄� \\ =鈭�n(n+1)鉄�n\left|a^{+}\right|n-1鉄� \\ =n(n+1)鉄�n鈭�n鉄� \\ =n(n+1){未}_{n}n \end{gathered}$$

Solve further

$$\begin{gathered} =n(n+1)鉄�n\left|a{a}^{+}{a}^{+}a\right|n鉄� \\ =\sqrt{n}鉄�n\left|a{a}^{+}{a}^{+}\right|n-1鉄� \\ =n鉄�n\left|a{a}^{+}\right|n鉄� \\ =n\sqrt{(n+1)}鉄�n\left|a\right|n+1鉄� \\ =n(n+1)鉄�n鈭�n鉄� \\ =n(n+1){未}_{n}n \\ =n(n+1) \\ =(n+1)鉄�n\left|a{a}^{+}\right|n鉄� \end{gathered}$$

Solve further

$$\begin{gathered} =(n+1)\sqrt{(n+1)}鉄�n\left|a\right|n+1鉄� \\ =(n+1{)}^2鉄�n鈭�n鉄� \\ =(n+1{)}^2{未}_{nn} \\ =(n+1{)}^2鉄�|aa{a}^{+}{a}^{+}|n鉄� \\ =\sqrt{(n+1)}鉄�n|aa{a}^{+}|n+1鉄� \\ =\sqrt{(n+1)}\sqrt{(n+2)}鉄�n|aa|n+2鉄� \\ =(n+2\left)\sqrt{(n+1}\right)鉄�n\left|a\right|n+1鉄� \\ =(n+1)(n+2)鉄�n鈭�n鉄� \\ =(n+1)(n+2){未}_{nn} \\ =(n+1)(n+2) \end{gathered}$$

Then,

$$\begin{gathered} 鉄�{\hat{x}}^4鉄�=\frac{{鈩�}^2}{4m^2{蝇}^2}\left[n(n-1)+n^2+n(n+1)+n(n+1)+(n+1{)}^2+(n+1\left)\right(n+2)\right] \\ =\frac{{鈩�}^2}{4m^2{蝇}^2}\left[n^2-n+n^2+n^2+n+n^2+n+n^2+2n+1+n^2+2n+n+2\right] \\ =\frac{{鈩�}^2}{4m^2{蝇}^2}[6n^2+6n+3] \end{gathered}$$

Now we can calculate the correction in the energy levels:

$$\begin{gathered} E_r^1=\frac{-1}{2m{c}^2}\left[\frac{1}{4}{m}^2{蝇}^4鈰�\frac{{鈩�}^2}{4m^2{蝇}^2}(6n^2+6n+3)\right] \\ =\frac{-{鈩�}^2{蝇}^2}{32m{c}^2}(6n^2+6n+3) \end{gathered}$$

Then, the lowest-order relativistic correction to the energy levels is

$E_r^1=\frac{-3{鈩�}^2{蝇}^2}{32m{c}^2}(2n^2+2n+1)$