The expectation value of p² of a particle in a cubic box of side ℓ having the wave function Ψnxnynz (x, y, z) =\(\rm \left( {\frac{2}{\ell }} \right)^{3/2} sin \frac{2 \pi x}{ℓ}sin \frac{3 \pi y}{ℓ}sin \frac{2 \pi z}{ℓ}\), is

Concept:

The average value of momentum operator:

For a 1D box, The momentum operator is represented by, 

\(\widehat P = - i\hbar \widehat {{d \over {dx}}}\).

Thus, \({\widehat P^2} = \left[ {{{\left( { - i\hbar } \right)}^2}{{\widehat {\left( {{\partial \over {\partial x}}} \right)}}^2}} \right]\) 

For a particle in ID box from 0 to ℓ, the wave function is \(\psi = \sqrt {{2 \over ℓ}} \sin \left( {{{n\pi x} \over ℓ}} \right)\).

For a particle in a 1D box of side length ℓ, the average value or expectation value of P² is

\(\left\langle {{{\widehat P}^2}} \right\rangle = \int\limits_0^ℓ {{\Psi ^*}{{\widehat P}^2}\Psi dx} \)

\( = \int\limits_0^A {{\Psi ^*}\left[ {{{\left( {{{\left( - i\hbar \right)}}{\partial \over {\partial x}}} \right)}^2}} \right]\Psi dx} \)

\( = \int\limits_0^A {{{\left( {\sqrt {{2 \over ℓ}} \sin \left( {{{n\pi x} \over ℓ}} \right)} \right)}^*}\left[ {{{\left( {{{\left( { - i\hbar } \right)}}{\partial \over {\partial x}}} \right)}^2}} \right]\left( {\sqrt {{2 \over ℓ}} \sin \left( {{{n\pi x} \over ℓ}} \right)} \right)dx} \)

\( = - {2 \over ℓ}{\hbar ^2}{{n\pi } \over ℓ}\int\limits_0^ℓ{\sin \left( {{{n\pi x} \over ℓ}} \right)\left( {{\partial \over {\partial x}}} \right)\cos \left( {{{n\pi x} \over ℓ}} \right)dx} \)

\( = {2 \over ℓ}{\hbar ^2}{\left( {{{n\pi } \over ℓ}} \right)^2}\int\limits_0^ℓ{\sin \left( {{{n\pi x} \over ℓ}} \right)\sin \left( {{{n\pi x} \over ℓ}} \right)dx} \)

\( = {{{\hbar ^2}} \over ℓ}{\left( {{{n\pi } \over ℓ}} \right)^2}\int\limits_0^ℓ{2{{\sin }^2}\left( {{{n\pi x} \over ℓ}} \right)dx} \)

\( = {{{\hbar ^2}} \over ℓ}{\left( {{{n\pi } \over ℓ}} \right)^2}\int\limits_0^ℓ {\left[ {1 - \cos \left( {{{2n\pi x} \over ℓ}} \right)} \right]dx} \)

\( = {{{\hbar ^2}} \over ℓ}{\left( {{{n\pi } \over ℓ}} \right)^2}\left[ {\int\limits_0^ℓ {dx} - \int\limits_0^ℓ {\cos \left( {{{2n\pi x} \over ℓ}} \right)dx} } \right]\)

\( = {{{n^2}{\pi ^2}{h^2}} \over {4{\pi ^2}{ℓ^3}}}\left[ {ℓ - 0} \right]\)

\( = {{{n^2}{h^2}} \over {4{ℓ^2}}}\)

Explanation:

• For a particle in a 1D-box, the average value or expectation value of P² is 

\(\left\langle {{{\widehat P}^2}} \right\rangle \) = \({{{n^2}{h^2}} \over {4{ℓ^2}}}\)

• The particle having a particular value of energy in the excited state may have several different stationary states or wavefunctions. If so, these states and energy eigenvalues are said to be degenerate.
• For a particle in a 3D box, with the wave function Ψnxnynz (x, y, z) =\(\rm \left( {\frac{2}{\ell }} \right)^{3/2} sin \frac{2 \pi x}{ℓ}sin \frac{3 \pi y}{ℓ}sin \frac{2 \pi z}{ℓ}\), the value of nx, ny, and nz are 2,3 and 2 respectively. 
• Similarly, for a particle in a 3D box with side length ℓ, the average value or expectation value of P² is \(\left\langle {{{\widehat P}^2}} \right\rangle \) 

= \(\left( {{n_x}^2 + {n_y}^2 + {n_z}^2} \right){{{h^2}} \over {4{A^2}}} \) 

• For a particle in a cubic box of side ℓ having the wave function 

​Ψnxnynz (x, y, z) =\(\rm \left( {\frac{2}{\ell }} \right)^{3/2} sin \frac{2 \pi x}{ℓ}sin \frac{3 \pi y}{ℓ}sin \frac{2 \pi z}{ℓ}\),

the value of n_x, n_y, and n_z are 2,3 and 2 respectively.

Conclusion:

Hence, the expectation value of P² of a particle in a cubic box of side ℓ is

 \(\left( {{2^2} + {3^2} + {2^2}} \right){{{h^2}} \over {4{l^2}}} = {{17{h^2}} \over {4{l^2}}}\)