Intro to PDE's, Schrodinger's Equation

Aryth · #1 $Old$ November 5th, 2009, 06:57 PM

(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?

Danny · #2 $Old$ November 6th, 2009, 09:50 AM

Quote:

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 $\psi(x)$ .