So starting from the time dependent schrodinger equation I perform separation of variables and obtain a time and spatial part. The spatial part is in effect the time independent schrodinger equation.
Since we are dealing with a free particle I can take the time independent equation, set V = 0 and solve.
I can do this successfully to obtain :
[itex]Ae^{+i\sqrt{{2mE}/{\hbar^{2}}}x}+Be^{-i\sqrt{{2mE}/{\hbar^{2}}}x}[/itex]
My lecturer has a small section titled :
So when I follow through and solve for the spatial part of his final solution I obtain :
[itex]A'e^{+ip/\hbar}+B'e^{-ip/\hbar}[/itex]
He seems to have conveniently ignored/left out one half of the above solution. Why ?
I am also aware that [itex]e^{-{iEt}/{\hbar}}[/itex] is simply the solution the time dependent part of the equation so my only issue is why he has left out a certain bit.
Since we are dealing with a free particle I can take the time independent equation, set V = 0 and solve.
I can do this successfully to obtain :
[itex]Ae^{+i\sqrt{{2mE}/{\hbar^{2}}}x}+Be^{-i\sqrt{{2mE}/{\hbar^{2}}}x}[/itex]
My lecturer has a small section titled :
Quote:
>Solving for the Free Schrodinger Equation [itex]V=0[/itex] [itex]\frac{\hbar^{2}}{2m}\frac{\partial^2\psi}{\partial x^2}+E\psi=0[/itex] [itex]E=\frac{p^2}{2m}[/itex] [itex]\psi=Ce^{-{iEt}/{\hbar}+{ipr}/{\hbar}}[/itex] This is the solution to the free TISE and TDSE. |
So when I follow through and solve for the spatial part of his final solution I obtain :
[itex]A'e^{+ip/\hbar}+B'e^{-ip/\hbar}[/itex]
He seems to have conveniently ignored/left out one half of the above solution. Why ?
I am also aware that [itex]e^{-{iEt}/{\hbar}}[/itex] is simply the solution the time dependent part of the equation so my only issue is why he has left out a certain bit.
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