Help regarding normalization of wave functions

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Hi, i need some help regarding normalization of a wave function, i feel it is a very simple problem, but i am having a hard time figuring it out. I would really appreciate it if anybody could help me out a bit regarding this.

I need to normalize the following wavefunctions by figuring out the constant B.

(1) ψ(x) = Bexp(ikx) where ψ(x) is non‐zero between x=0 and x=L ;
(2) ψ(x) = Bexp(−kx) where ψ(x) is non‐zero between x=0 and x=∞

Thanks a lot in advance.

 

[jtbell]

Do you know what it means for a function to be normalized? What equation or property does a normalized function have to satisfy?

 

[dacruick]

|ψ(x)|² = 1

So legend square both sides, set |ψ(x)|² equal to one. then solve for B

 

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Thanks a lot dacruick.

@jtbell... actually i have some vague idea, i am still in the process of coming to terms with quantum mechanics (i don't have a physics background :-( )

 

[Avodyne]

|ψ(x)|² = 1

Wrong.

 

[jtbell]

|ψ(x)|² = 1

No, you also have to integrate. In general:

$$\int^{+\infty}_{-\infty}{|\psi(x)|^2 dx}<br /> = \int^{+\infty}_{-\infty}{\psi^*(x)\psi(x) dx} = 1$$

In this case, one actually has to integrate only over the region where $\psi(x)$ is non-zero.

 

[Halcyon-on]

No, you also have to integrate. In general:

$$\int^{+\infty}_{-\infty}{|\psi(x)|^2 dx}<br /> = \int^{+\infty}_{-\infty}{\psi^*(x)\psi(x) dx} = 1$$

In this case, one actually has to integrate only over the region where $\psi(x)$ is non-zero.

If $$\psi(x)$$ is a period wave with period L or it is in a box

$$\int^{+L}_{0}{|\psi(x)|^2 dx}<br /> = \int^{+L}_{0}{\psi^*(x)\psi(x) dx} = 1$$

 

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Thanks a lot for your answers. I have one doubt.

How do i go about the second function i.e Bexp(−kx)? Do i replace "-k" by "i²k" and proceed?

 

[Truecrimson]

The wave function is real so $$\psi=\psi^*=Be^{-kx}$$. Just integrate as it is and you'll be fine. (Replacing $$-k\to i^2k$$ will also give the correct answer because it actually does nothing. You don't have to do that.)

 

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Thanks a lot to all of you guys for your help. So i solved them and got the following results. Could anyone please let me know if they are correct?
For number 1 : B = 1/(√L)
For number 2 : B = √(2k)

 

[Truecrimson]

Yes, they are correct.

 

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Thanks a lot