Question

Consider the one-dimensional Time-Dependent Schrodinger Equation: \begin{equation} -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi(x,t)}{\partial x^2} + V(x)\Psi(x,t) - i\hbar \frac{\partial \Psi(x,t)}{\partial t} = 0 \end{equation} Insert the stationary state, $\Psi(x,t) = \psi(x)e^{-i\omega t}$ into the TDSE and show that the result is the Time-Independent Schrodinger Equation (TISE): \begin{equation} -\frac{\hbar^2}{2m}\frac{d^2 \psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x) \end{equation} where $E = \hbar \omega$.

          Consider the one-dimensional Time-Dependent Schrodinger Equation:
\begin{equation}
-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi(x,t)}{\partial x^2} + V(x)\Psi(x,t) - i\hbar \frac{\partial \Psi(x,t)}{\partial t} = 0
\end{equation}
Insert the stationary state, $\Psi(x,t) = \psi(x)e^{-i\omega t}$ into the TDSE and show that the
result is the Time-Independent Schrodinger Equation (TISE):
\begin{equation}
-\frac{\hbar^2}{2m}\frac{d^2 \psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)
\end{equation}
where $E = \hbar \omega$.

Consider the one-dimensional Time-Dependent Schrodinger Equation:

-(ħ^2)/(2m)(∂^2 Ψ(x,t))/(∂ x^2) + V(x)Ψ(x,t) - iħ(∂Ψ(x,t))/(∂ t) = 0

Insert the stationary state, Ψ(x,t) = ψ(x)e^-iω t into the TDSE and show that the
result is the Time-Independent Schrodinger Equation (TISE):

-(ħ^2)/(2m)(d^2 ψ(x))/(dx^2) + V(x)ψ(x) = Eψ(x)

where E = ħω.

Added by Crystal C.

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