Solving the Schrodinger Equation for A and l

[Tranquillity]

I have the following Schrödinger equation:

i* (h-bar) * partial derivative of ψ(x,t) w.r.t time
=
[(m*w^2 / 2) * x^2 * ψ(x,t) ] - (1/2m) * (h-bar)^2 * (laplacian of ψ(x,t))]

m>0 is the mass
w is a positive constant

Assume that the ground state (normalizable energy eigenfunction) with the lowest possible energy E(0) is of the form

ψ(x ,t) = A * exp ((-i * E(0)*t/ (h-bar) ) - l * x^2)

A, l are constants

Use the equation to find A and l.


My try:


I know that the Scrodinger eqn can be reduced to a time - independent form which in my case would be


E * ψ (E) = { (-(h-bar)^2 / 2m )* laplacian + (m*w^2 / 2) * x^2} * ψ (E)

Then I am not sure how to proceed.

For the normalization constant I know that the integral from minus infinity to infinity of ψ(x,t)^2 = 1 by Born interpretation for the probability density.

Any help would be greatly appreciated!

Thank you!

 

[vela]

Plug the function into the Schrödinger equation. What do you get?

 

[Tranquillity]

After substituting I am getting the relation:

E(0) = ( (-l * (h - bar)^2) / m ) + 4 * l^2 * x^2 + ( (m * w^2) / 2 ) * x^2

But after that how do I proceed?

 

[vela]

E can't depend on x. E is a constant.

You need to show your work. We don't know what you did and where your mistake is if all you do is post your final result.

 

[Tranquillity]

These are my workings, any reply will be greatly appreciated!Wrong workings! I have to use the time dependent Schrödinger equation!

 

[Tranquillity]

Since I am given something like a trial solution and I have to find the constant λ I worked using the time - dependent Schrödinger equation but I reach a point I don't know how to continue!


Can someone please review my work and guide me because I am really stuck at the problem?


Thank you very much!

 

[vela]

Something is wrong with your work since the units don't work out. You need to find and fix your algebra mistakes. Try checking the units after each step until you find they don't work.

You should end up with some coefficient of x² that depends on $\lambda$. For the right choice of $\lambda$, that coefficient will equal 0, so you're just left with a constant.

 

[Tranquillity]

In my last equation if you see there is a coefficient of x^2 based on λ. I did at least three times the same calculation by hand or by Maple reaching the same equation.

Have you seen the second attachment and you find something wrong? (because my first attachment is not the way)

I don't know what goes wrong, if you can work out the calculations too I would be greatly appreciated!

In my last equation I find the coefficient of x^2 to be dependent on lamda.

 

[Tranquillity]

I did again the calculations finding the exact same answer (as the attachment "Final Workings")

By rearranging I get E - ((E^2)/2m) - (λ * (h-bar)^2)/m) + (2 * λ^2 * x^2 * ((h-bar)^2))/m - (m*w^2 * x^2)/2 = 0


Then as you said E is constant so the terms involving x^2 must disappear.

So (2*λ^2 * (h-bar)^2) / m = m * w^2 / 2

and by a final rearrangement I get λ = (m * w) / (2 * h-bar)


And indeed λ is a constant.


What do you think?


The only thing I realized is that my equation is of the Quantum Oscillator which by some searching I did, my λ is correct.


The "problem" I find is that E(0) of Quantum Oscillator equals with h-bar * w / 2 and if I substitute E(0) in the equation I deduced, it is not satisfied.


Not sure what goes wrong, I really did the calculations many many times resulting in the same thing!

 

[vela]

How can you subtract $E^2/2m$ from E? The units don't match.

 

[Tranquillity]

Ok I found my mistake, all this time I was using a wrong laplacian! My laplacian should be only the second partial derivative of the wavefunction w.r.t x and it shouldn't have involved the second partial derivative of the wavefunction w.r.t time!Thanks again for the help!

 

[vela]

Glad you figured it out!