Normalizing a wave function problem

[Gatsby88]

Homework Statement

Normalize the wave function

ψ(x,0) = C^1/4 * e^a(x2)-ikx a and k are positive real constants

Homework Equations

∫|ψ|²dx = 1

The Attempt at a Solution

Now, my maths is a little weak, so I'm struggling a little bit here.

The constant is easy to deal with in all aspects of this problem, so that doesn't worry me, and I've ignored that below to make it easier to read.

When I'm squaring the e function, do I just square it or do I multiply by its complement?

If I use complements, the i bit goes away leaving e^-2ax2.

If I square, I get e^-2ax2-2ikx This can then be rewritten in the form of cos and sin which allows me to say that the sin function is odd so its integral is 0 and can be ignored (removing the imaginary part again), but it still leaves me with e^-2ax2cos(2kx)

The question goes on to say that I should change the variable of integration and use a standard integral, and I can't see a standard integral looking like either of my avenues that I've pursued above.

 

[jtbell]

When I am squaring the e function, do I just square it or do I multiply by its complement?

By definition, $|\Psi|^2 = \Psi^*\Psi$. That is, the complex conjugate of $\Psi$ times $\Psi$ itself.

I can't see a standard integral looking like either of my avenues that I've pursued above.

Speak the magic words "Gaussian integral" to Google and you will find what you're looking for.

 

[Gatsby88]

If I just had the e^-2ax2 term, I can see I would need to use the identity

∫e^-x2 dx = pi

But I don't know how to change that given this term is multiplied by cos(2kx) in my problem

I can't find any gaussian integral that relates e and cos.

Im sure its just a manipulation problem but I am not too good at manipulating integrals and this is where I am finding myself stuck since I am not entirely sure how to go about it.

 

[atyy]

By "complement" do you mean "complex conjugate", in which "i" is replaced by "-i"?

 

[Gatsby88]

Yes. In this case we replace the (-i) with i which when squared and ignoring the constant ends up giving me e^-2ax2cos(2kx) ?

 

[atyy]

Yes. In this case we replace the (-i) with i which when squared and ignoring the constant ends up giving me e^-2ax2cos(2kx) ?

Don't square it - multiply it with the original expression.

In quantum mechanics, the "squaring" notation means to multiply an expression with its "complement" or "complex conjugate".

 

[Gatsby88]

ψ(x,0) = C^1/4 * e^ax2-ikx a and k are +ve real constants

So C^1/4 * e^-ax2-ikx

= C^1/2 * e^-ax2-ikx * e^-ax2+ikx

= C^1/2 * e^-2ax2

Now ∫e^-2ax2 dx = [1/√(2a)]pi using the gaussian integral here

The constant was (2a/pi) so the whole thing becomes

√(2a/pi) * √(1/2a) * pi

The √2a cancels and we're left with pi/√pi which is √pi

Is this sufficient to prove the wave function is normalized? The answer is finite but is not 1. You can make it 1 by multiplying by 1/√pi or does this mean I've made an error somewhere?

 

[atyy]

I'm terrible at algebra, but shouldn't the Gaussian integral be √(pi/(2a))?

 

[Gatsby88]

Absolutely. Id forgotten that it was √pi and not merely pi

Carrying this through gives √pi/√pi which is 1 as required!

Thank you for the help, I feel I understand where I was going wrong much better now.

 

[atyy]

I was about to say just normalize it, then I saw they'd chosen a nice value for C.

 

[Gatsby88]

How do I go about manipulating gaussian (standard) integrals?

From this question I can say that

∫e^-2ax2 dx = √(pi/2a)

I have a follow on question that, after doing some manipulation, means I need to find

∫x²e^-2ax2 dx

 

[Gatsby88]

FWIW, looking here gives me a lot of help and I think that

∫x²e^-2ax2 dx = 1/2 * √[pi/(2a)^3]

Its 1/2 and not 1/4 because the standard integral on wiki goes from 0 to inf and we need the area from -inf to inf. Since I am dealing with an even function I can just double the area.

Grateful though I am for wiki giving me the answer, the question only mentions the standard integral of e^-2ax2 so there must be a way of getting to the standard integral for ∫x²e^-2ax2 dx that doesn't require the need for searching.

 

[atyy]

I think a standard trick is to differentiate with respect to "a" .

BTW, is this a homework question?

 

[jtbell]

there must be a way of getting to the standard integral for ∫x²e^-2ax2 dx that doesn't require the need for searching.

Try the "See Also" links at the bottom of the Wikipedia page.

 

[jtbell]

The answer is finite but is not 1. You can make it 1 by multiplying by 1/√pi or does this mean I've made an error somewhere?

You left out the C at the beginning of your original expression for $\Psi$. Usually, when we say, "normalize this wave function", there's an arbitrary constant in front (your C), and you want to find the value of C that makes the whole thing normalized.

At least, I assume your C is arbitrary. Does your source specify its value to begin with?

 

[Gatsby88]

Try the "See Also" links at the bottom of the Wikipedia page.

Awesome. Will do, Thanks.

BTW, is this a homework question?

Its a set question, but one that doesn't impact upon my grade at all.

 

[jtbell]

If you find yourself searching for integrals, trig identities, etc., often, you might want to invest in this book:

https://www.amazon.com/dp/1439835489/?tag=pfamazon01-20

I've used various editions of it since I was an undergraduate forty years ago. I always have it handy when I'm slogging through a derivation: quantum, E&M, etc.

 

[micromass]

Its a set question, but one that doesn't impact upon my grade at all.

OK, but such questions should be asked in the homework forum. I'll move it for you.