Thread: Intro to PDE's, Schrodinger's Equation
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Old November 6th, 2009, 08:50 AM
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Originally Posted by Aryth View Post
(a) Using the separation of variables technique, and letting the separation constant be denoted E (which turns out to be the total energy of the particle), show that the resulting differential equation that is independent of time—the so-called time-independent Schroedinger equation (TISE)—has the (1-D) form

-\frac{\hbar}{2m}\frac{d^2\psi}{dx^2} + U(x)\psi(x) = E\psi{x}

where we have set \Psi(x,t) = \psi(x) T(t).

This I have done this already... And I've narrowed down to these two differential equations:

The one above

i\hbar \frac{dT}{dt} = ET(t)

(b) Also, show that the total wavefunction in this case has the form

\Psi(x,t) = \psi(x) e^{-i\frac{E}{\hbar}t}

Here's what I did:

i\hbar \frac{dT}{dt} = ET(t)

\frac{dT}{T} = \frac{E}{i\hbar}dt

ln|T| = \frac{Et}{i\hbar} + C

T = e^{\frac{Et}{i\hbar} + C}

T = Ae^{\frac{Et}{i\hbar}}

This looks only similar to the actual solution for T(t)... What am I doing wrong?
Note \frac{1}{i} = - i and absorb A into \psi(x).
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