Homework Help: Wave function

1. Jun 28, 2011

athrun200

1. The problem statement, all variables and given/known data

I don't know how to prove b.)

2. Relevant equations

3. The attempt at a solution

What can I do after writing these 4 eqtn?

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2. Jun 28, 2011

Clever-Name

You're almost there. What do you know about a linear combination of solutions to a differential equation? What does that say about each independent solution?

3. Jun 28, 2011

athrun200

In fact that is the hint.
So can you also explain the hint?

And I still don't know how to get the answer:(

4. Jun 28, 2011

Clever-Name

I'm asking it in a different way to get you to think about it. Do you know what a linear combination is? (If not then say so and I'll explain it). Have you taken a course in differential equations?

If both elements in the linear combination are individually solutions to the differential equation then is it also true that the two elements combined in a linear combination is then a solution?

5. Jun 28, 2011

athrun200

Yes, I have taken courses about that and I know how to solve first and second linear differential eqtn.
However, all I know is how to calculate. I am not fully understand the meaning behind it.

Do you mean that I need to prove the solution for fourth eqtn is real.

i.e. $i(\psi-\psi*)$ is real?

6. Jun 28, 2011

ideasrule

A differential equation is linear if none of the terms are being squared, cubed, square rooted, etc. Schrodinger's equation is linear because the wavefunction isn't being raised to any power, and neither is its second derivative.

For a linear system of equations (not necessarily differential equations), if you know that A and B are both solutions, all linear combinations of A and B must also be solutions. Here, you need to prove that if ψ is a solution to Schrodinger's equation, so is ψ*. Then you'll know that ψ+ψ* and i(ψ−ψ∗) are also solutions to Schrodinger's equation. But these are both real, which completes the proof.

7. Jun 29, 2011

athrun200

I wonder if i(ψ−ψ∗) is real, because it has a " i " in front of it.

8. Jun 29, 2011

vela

Staff Emeritus
Let $\psi(x) = u(x) + i v(x)$ where u(x) and v(x) are real functions. Then $\psi^*(x) = u(x) - i v(x)$.

What are $\psi(x)+\psi^*(x)$ and $i(\psi(x)-\psi^*(x))$ equal to?

9. Jun 29, 2011

athrun200

Oh! Thanks so much!!
Why I am so stupid that I didn't try to let $\psi^*(x) = u(x) - i v(x)$.

10. Jun 29, 2011

athrun200

Wait a minute, it is easy to solve because of the hint.
How about without the hint? How can I get the fourth eqtn?

Can I obtain it like this?
By mutiplying $i$ on eqtn 1 and 2, implys that $i\psi(x)$ and $i\psi(x)^*$ are soln to the eqtn.

By subtracting them get the fourth eqtn.

Am I right?

11. Jun 29, 2011

Clever-Name

Here's my take on it:

Given

$$-\frac{{\hbar}^{2}}{2m}\frac{d^{2}{\psi}}{dx^{2}} + V{\psi} = E{\psi}$$

Taking the complex conjugate:

$$-\frac{{\hbar}^{2}}{2m}\frac{d^{2}{\psi}^{*}}{dx^{2}} + V{\psi}^{*} = E{\psi}^{*}$$

It's important to note that both V and E are real.

If we let ${\psi}_{1}$ and ${\psi}_{2}$ satisfy the first equation then so does some linear combination of them ${\psi}_{3}$ where:
$${\psi}_{3} = c_{1}{\psi}_{1} + c_{2}{\psi}_{2}$$

If we were to sub ${\psi}_{3}$ into the first equation we would expand out, rearrange blah blah blah simplify back down to the first equation again just with ${\psi}_{3}$ in place of ${\psi}$

So there we have proved that a linear combination will satisfy the equation. We also know that both ${\psi}$ and ${\psi}^{*}$ are independent solutions.

The linear combinations:
$${\psi} + {\psi}^{*}$$
and
$$i({\psi} - {\psi}^{*})$$
are solutions and are also real.

Thus we can conclude that from any complex solution we can construct two real solutions.

We can write:

$${\psi} = \frac{1}{2}[({\psi} + {\psi}^{*}) -i( i({\psi} - {\psi}^{*}))]$$
which we can simplify down to just ${\psi}$ we have thus proven that any ${\psi}$ can be written as a linear combination of two real solutions.