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dotman
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Hi, I've got a problem with the following problem. This is 1.8 out of Griffiths QM text, and was previously covered on this forum for another user in https://www.physicsforums.com/showthread.php?t=152775", although that thread doesn't address my problem.
1. Suppose a constant potential energy,Vo, independent of x and t is added to a particle's potential energy. Show that this adds a time-dependent phase factor, [tex] e^{-iV_ot/\hbar}[/tex].
My attempt at the solution was, as stated in the other thread, to sub in [tex] \psi(x,t) e^{-iV_0 t/\hbar} [/tex] and see if that solves the schrodinger eqn with the constant potential V0 added in:
[tex]i\hbar\frac{d\Psi}{dt} = \frac{-\hbar^2}{2m}\frac{d^2\Psi}{dx^2} + V(x)\Psi[/tex]
[tex]\psi=\psi e^{-iV_{0}t/\hbar}[/tex]
[tex]\Rightarrow\frac{\partial\psi}{\partial t}=\frac{\partial\psi}{\partial t}e^{-iV_{0}t/\hbar}-\frac{iV_{0}}{\hbar}\psi e^{-iV_{0}t/\hbar}[/tex]
[tex]\frac{\partial\psi}{\partial x}=\frac{\partial\psi}{\partial x}e^{-iV_{0}t/\hbar}[/tex]
[tex]\frac{\partial^{2}\psi}{\partial x^{2}}=\frac{\partial^{2}\psi}{\partial x^{2}}e^{-iV_{0}t/\hbar}[/tex]
[tex]i\hbar\left[\frac{\partial\psi}{\partial t}e^{-iV_{0}t/\hbar}-\frac{iV_{0}}{\hbar}\psi e^{-iV_{0}t/\hbar}\right]=\frac{-\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}e^{-iV_{0}t/\hbar}+V\psi e^{-iV_{0}t/\hbar}[/tex]
[tex]\Rightarrow i\hbar\left[-\frac{iV_{0}}{\hbar}\psi e^{-iV_{0}t/\hbar}\right]+i\hbar\frac{\partial\psi}{\partial t}e^{-iV_{0}t/\hbar}=\frac{-\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}e^{-iV_{0}t/\hbar}+V\psi e^{-iV_{0}t/\hbar}[/tex]
[tex]\Rightarrow i\hbar\left[-\frac{iV_{0}}{\hbar}\psi\right]+i\hbar\frac{\partial\psi}{\partial t}=\frac{-\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}+V\psi[/tex]
[tex]\Rightarrow-\frac{i^{2}V_{0}\hbar}{\hbar}\psi+i\hbar\frac{\partial\psi}{\partial t}=\frac{-\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}+V\psi[/tex]
[tex]\Rightarrow V_{0}\psi+i\hbar\frac{\partial\psi}{\partial t}=\frac{-\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}+V\psi[/tex]
[tex]\Rightarrow i\hbar\frac{\partial\psi}{\partial t}=\frac{-\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}+V\psi-V_{0}\psi[/tex]
[tex]\Rightarrow i\hbar\frac{\partial\psi}{\partial t}=\frac{-\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}+\left(V-V_{0}\right)\psi[/tex]
But we're supposed to have added the constant potential in, so it should look like:
[tex]i\hbar\frac{\partial\psi}{\partial t}=\frac{-\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}+\left(V+V_{0}\right)\psi[/tex]
Have I made a mistake somewhere? Am I doing this wrong? Any help or advice would be appreciated.
Thanks!
1. Suppose a constant potential energy,Vo, independent of x and t is added to a particle's potential energy. Show that this adds a time-dependent phase factor, [tex] e^{-iV_ot/\hbar}[/tex].
My attempt at the solution was, as stated in the other thread, to sub in [tex] \psi(x,t) e^{-iV_0 t/\hbar} [/tex] and see if that solves the schrodinger eqn with the constant potential V0 added in:
[tex]i\hbar\frac{d\Psi}{dt} = \frac{-\hbar^2}{2m}\frac{d^2\Psi}{dx^2} + V(x)\Psi[/tex]
[tex]\psi=\psi e^{-iV_{0}t/\hbar}[/tex]
[tex]\Rightarrow\frac{\partial\psi}{\partial t}=\frac{\partial\psi}{\partial t}e^{-iV_{0}t/\hbar}-\frac{iV_{0}}{\hbar}\psi e^{-iV_{0}t/\hbar}[/tex]
[tex]\frac{\partial\psi}{\partial x}=\frac{\partial\psi}{\partial x}e^{-iV_{0}t/\hbar}[/tex]
[tex]\frac{\partial^{2}\psi}{\partial x^{2}}=\frac{\partial^{2}\psi}{\partial x^{2}}e^{-iV_{0}t/\hbar}[/tex]
[tex]i\hbar\left[\frac{\partial\psi}{\partial t}e^{-iV_{0}t/\hbar}-\frac{iV_{0}}{\hbar}\psi e^{-iV_{0}t/\hbar}\right]=\frac{-\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}e^{-iV_{0}t/\hbar}+V\psi e^{-iV_{0}t/\hbar}[/tex]
[tex]\Rightarrow i\hbar\left[-\frac{iV_{0}}{\hbar}\psi e^{-iV_{0}t/\hbar}\right]+i\hbar\frac{\partial\psi}{\partial t}e^{-iV_{0}t/\hbar}=\frac{-\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}e^{-iV_{0}t/\hbar}+V\psi e^{-iV_{0}t/\hbar}[/tex]
[tex]\Rightarrow i\hbar\left[-\frac{iV_{0}}{\hbar}\psi\right]+i\hbar\frac{\partial\psi}{\partial t}=\frac{-\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}+V\psi[/tex]
[tex]\Rightarrow-\frac{i^{2}V_{0}\hbar}{\hbar}\psi+i\hbar\frac{\partial\psi}{\partial t}=\frac{-\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}+V\psi[/tex]
[tex]\Rightarrow V_{0}\psi+i\hbar\frac{\partial\psi}{\partial t}=\frac{-\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}+V\psi[/tex]
[tex]\Rightarrow i\hbar\frac{\partial\psi}{\partial t}=\frac{-\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}+V\psi-V_{0}\psi[/tex]
[tex]\Rightarrow i\hbar\frac{\partial\psi}{\partial t}=\frac{-\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}+\left(V-V_{0}\right)\psi[/tex]
But we're supposed to have added the constant potential in, so it should look like:
[tex]i\hbar\frac{\partial\psi}{\partial t}=\frac{-\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}+\left(V+V_{0}\right)\psi[/tex]
Have I made a mistake somewhere? Am I doing this wrong? Any help or advice would be appreciated.
Thanks!
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