91影视

The Schr枚dinger equation is given by,

$\mathbf{i}\sout{\mathbf{h}}\frac{\mathbf{鈭�}\mathbf{蠄}}{\mathbf{鈭�}\mathbf{t}}\mathbf{=}\frac{\mathbf{-}\sout{\mathbf{h}}}{\mathbf{2}\mathbf{m}}\frac{{\mathbf{鈭�}}^{\mathbf{2}}\mathbf{蠄}}{\mathbf{鈭�}{\mathbf{X}}^{\mathbf{2}}}\mathbf{+}\mathbf{痴蠄}$ 鈥︹�� (1)

This equation is without a constant V₀.

If a constant is added to the potential energy, the Schr枚dingerequation will be,

$\mathrm{i}\sout{\mathrm{h}}\frac{鈭�\mathrm{蠄}}{鈭�\mathrm{t}}=\frac{-\sout{\mathrm{h}}}{2\mathrm{m}}\frac{{鈭�}^2\mathrm{蠄}}{鈭�{\mathrm{X}}^2}+(\mathrm{V}+{\mathrm{V}}_0)\mathrm{蠄}$ 鈥︹�� (2)

A time-dependent phase factor will be picked up by the wave function that is ${\mathrm{e}}^{-{\mathrm{N}}_0\mathrm{t}/\sout{\mathrm{h}}}$. For doing the calculation a new wave function $蠒$will be taken that is equal to the old wavefunction which is multiplied with the phase factor.

$\mathrm{桅}={\mathrm{蠄别}}^{-{\mathrm{iv}}_0\mathrm{t}/\sout{\mathrm{h}}}$

Now, equation (1) will be replaced with this new wave function (the originalSchr枚dingerequation before adding the constant to the potential energy). So, the equation will be:

$$\begin{gathered} i\sout{h}\frac{鈭�桅}{鈭�t}=\frac{-\sout{h}}{2m}\frac{{鈭�}^2桅}{鈭�X^2}+V桅 \\ i\sout{h}\frac{鈭�桅}{鈭�t}\left(蠄e^{-i{v}_{0}t/\sout{h}}\right)=\frac{-\sout{h}}{2m}\frac{{鈭�}^2}{鈭�X^2}\left(蠄e^{-i{v}_{0}t/\sout{h}}\right)+V\left(蠄e^{-i{v}_{0}t/\sout{h}}\right) \\ i\sout{h}\frac{鈭�蠄}{鈭�t}{e}^{-i{v}_{0}t/\sout{h}}+蠄V_0{e}^{-i{v}_{0}t/\sout{h}}=\frac{-\sout{h}}{2m}\frac{{鈭�}^2蠄}{鈭�X^2}{e}^{-i{v}_{0}t/\sout{h}}+V蠄e^{-i{v}_{0}t/\sout{h}} \\ i\sout{h}\frac{鈭�蠄}{鈭�t}=\frac{-\sout{h}}{2m}\frac{{鈭�}^2蠄}{鈭�X^2}+V蠄-蠄V_0 \\ i\sout{h}\frac{鈭�蠄}{鈭�t}=\frac{-\sout{h}}{2m}\frac{{鈭�}^2蠄}{鈭�X^2}+(V+V_0)蠄 \end{gathered}$$

The negative sign is absorbed into the constant in this case. The extra phase factor will also have no effect on the dynamical variables' expected values.

Because the dynamical variables do not depend on time (just on position), the extra phase component will have no effect on the dynamical variables' expected value.

Hence, it鈥檚 the same as the old potential energy.