Question about schroedinger equation

[hadi2003]

pliz help me how to solve the schroedinger equation from the time dependent to become the time independent?

thank you

 

[heardie]

You'll have to excuse any Latex errors I haven't quite worked out what Tex this board uses yet. anyway:
$$<br /> \frac{{ - \hbar ^2 }}<br /> {{2m}}\frac{{\partial ^2 \Psi (x,t)}}<br /> {{\partial x^2 }} + V(x)\Psi (x,t) = i\hbar \frac{{\partial \Psi (x,t)}}<br /> {{\delta t}}<br /> <br /> <br /> \]<br />$$

Assume seperable:
$$\Psi (x,t) = \psi (x)\phi (t)$$
Substitute:
$$<br /> <br /> \frac{{ - \hbar ^2 }}<br /> {{2m}}\frac{{\partial ^2 \psi (x)\phi (t)}}<br /> {{\partial x^2 }} + V(x)\Psi (x,t) = i\hbar \frac{{\partial \psi (x)\phi (t)}}<br /> {{\delta t}}<br /> <br />$$
Take terms out of x derivate that are t-dependent, and vice-versa - also divide by $$\psi (x)\phi (t)$$:
$$<br /> {{ - \hbar ^2 } \over {2m}}{1 \over {\psi (x)}}{{\partial ^2 \psi (x)} \over {\partial x^2 }} + V(x) = i\hbar {1 \over {\phi (t)}}{{\partial \phi (t)} \over {\delta t}}<br />$$

Set each side equal to a costant:
$$<br /> \hbar i{1 \over {\phi (t)}}{{d\phi (t)} \over {dt}} = E<br />$$
This yields:
$$<br /> \phi = Ae^{ - i\omega t} <br />$$
The other equation, becomes (upon rearranging):
$$\frac{{ - \hbar ^2 }}<br /> {{2m}}\frac{{d^2 \psi (x)}}<br /> {{dx^2 }} + V(x)\psi (X) = E\psi (x)<br />$$
There is a quick sketc. However, it is a worthwhile problem to do yourself - separation of variables is a very useful technique!

 

[maverick280857]

Basic Separation of Variables

HI

You have not mentioned the situation for which you want the solution. So I am assuming that you are interested in getting the standard one,

$$\nabla^2 \psi + \frac{8\pi^{2}m}{h^{2}} (E-V)\psi = 0$$

Here, E is the total particle energy, V is the potential energy, psi is the wavefunction for the particle, m is its mass, h is the Planck Constant.

Steps

We wish to write the wave equation for a particle, so generally, it is

$$\nabla^2 \psi = \frac{1}{v^{2}} \frac{\partial^2{\psi}}{\partial{t}^2}$$

This is the starting point for all our computations. Now, in order to separate the time dependent parts (containing t) and the time-independent parts (containing x,y,z) we perform a technique called Separation of Variables. It is more generally used to solve partial differential equations.

We write the total wavefunction as a product of two wavefunctions--one dependent only on spatial coordinates (x,y,z) and not on time and the other dependent only on time and not on (x,y,z).

$$\psi_{(xyzt)} = \psi_{(xyz)}g(t)$$

Now clearly g(t) must be solely time dependent, so for the sake of convenience, we usually write

$$g(t) = g_{0}e^{2 \pi i \nu t} \qquad \mbox{where i = \sqrt{-1}} \qquad \mbox{g_{0} = const.}$$

Note that g(t) is the most general complex function depending only on time.

Substitute the (assumed) expression for the total wavefunction (including g(t) as a product) in the wave equation. Note that the laplacian is a space operator only (contains only space coordinates) and so g(t) remains unaffected by it. After a little bit of algebraic manipulations which should now be clear, you get

$$\nabla^{2}\psi_{(xyz)} = -\frac{4 \pi^{2}\nu^{2}}{v^{2}}\psi_{(xyz)}$$

If the particle analog of the expression,

$$c = \lambda \nu$$

that is,

$$v = \lambda \nu = \frac{h\nu}{p}$$

is substituted for v in the above wave equation, it becomes

$$\nabla^{2}\psi_{(xyz)} = -\frac{4 \pi^{2}p^{2}}{h^{2}}\psi_{(xyz)}$$

The linear momentum of the particle p, is related to the kinetic energy T as

$$T = \frac{1}{2}mv^{2} = \frac{p^{2}}{2m}$$

Substituting for p into the equation and writing the kinetic energy T as (E-V) where E is the total energy of the particle and V is its potential energy, the wave equation becomes

$$\nabla^2 \psi + \frac{8\pi^{2}m}{h^{2}} (E-V)\psi = 0$$

(This psi depends only on space coordinates, the (xyz) subscripts dropped.)

This is the time independent form of the Schroedinger Wave Equation. The algorithm for obtaining it is stated thus:

1. Consider the general wave equation in terms of the velocity of the particle.
2. Write the total wavefunction as a product of two wavefunctions, one containing space coordinates only and the other containing time only. Simply to remove the time dependence.
3. Use De-broglie's Relation to introduce particle character.
4. Write T in terms of p.
5. Replace T by E-V.
6. Bingo!

Hope that helps...

Cheers
Vivek

 

[hadi2003]

thank you maverick280857

 

[hadi2003]

thank you heardie