Given that the commutator \(\left[\widehat{A}^2,\widehat{B}\right] =\left[\widehat{A},\widehat{B}\right]\widehat{A}+\widehat{A}\left[\widehat{A},\widehat{B}\right]\) the value of \(\left[x,[\widehat{p^2_x},x]\right]\) is

Concept:-

For linear and position momentum:

[x̂, p̂x] = iℏ   ....(1)

[p̂x, x̂] = -iℏ   ...(2)

[x̂n, p̂x] = nxn-1 [x, p̂x]   ....(3)

[p̂xn, x̂] = npxn-1[p̂x, x̂] .....(4)

Explanation:-

• Given that the commutator 

\(\left[\widehat{A}^2,\widehat{B}\right] =\left[\widehat{A},\widehat{B}\right]\widehat{A}+\widehat{A}\left[\widehat{A},\widehat{B}\right]\)

• \(\left[x,[\widehat{p^2_x},x]\right]\)

= \(\left[x, \left[\widehat{p_x},\widehat{x}\right]\widehat{p_x}+\widehat{p_x}\left[\widehat{p_x},\widehat{x}\right ]\right]\)...........(5)

• Now using equation (4) and (1),

[p̂x, x̂] = -iℏ 

• Now, from equation (5) we get,

\(\left[x, \left[\widehat{p_x},\widehat{x}\right]\widehat{p_x}+\widehat{p_x}\left[\widehat{p_x},\widehat{x}\right ]\right]\)

= \(\left[x, \left ( -i\hbar\right )\widehat{p_x}+\widehat{p_x}\left ( -i\hbar\right )\right]\)

= \(\left[x, -2i\hbar \widehat{p_x}\right]\)

= \( -2i\hbar \left[x, \widehat{p_x}\right]\)

= \( -2i\hbar \times i\hbar\)

= \( -2i^2\hbar^2 \)

= \( 2\hbar^2 \)

Conclusion:-

Hence, the value of \(\left[x,[\widehat{p^2_x},x]\right]\) is \( 2\hbar^2 \)