Normalization of a wavefunction problem

In summary, a homework equation is integral (psi*psi)dx=1, and the attempt at a solution is to integrate sin((n*pi*x)/L) where x is between 0 and L and n is a positive integer. Squaring the function will just confuse you, and the usual way to do it is to write\psi_n(x) = N \sin{\left(\frac{n\pi x}{L}\right)where N is the yet unknown normalization constant. Then you plug this wavefunction into1=\int_0^L \psi_n^*(x)\psi_n(x) dx
  • #1
hellomister
29
0

Homework Statement


Normalize sin ((n*pi*x)/L) where x is between 0 and L and n is a positive integer


Homework Equations


integral (psi*psi)dx=1
N^2 integral sin ((n*pi*x)/L)dx =1

I don't really understand if this integral is correct, what is the complex conjugate of the wavefunction?

Can i just integrate my wavefunction and say tht is psi*psi?


The Attempt at a Solution


N^2 integral sin ((n*pi*x)/L)dx =1

-N^2 Lcos ((n*pi*x/L))/n*pi=1
I just integrated the wavefunction and then solved for N
 
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  • #2
The complex conjugate of a real number or function is just that number/function itself. You should be squaring your function. You can't just integrate your function alone, it has to be multiplied by itself.
 
  • #3
Ohh, thanks! I was very confused.
So then I would square my function and then integrate and solve for N?
then my normalized wavefunction would be N times the original function?
 
  • #4
Yes. I don't know why you have a N squared though. There's no reason for N to be squared. N is just some as yet undetermined constant. Squaring it will just confuse you.
 
  • #5
The usual way it's done is you write

[tex]\psi_n(x) = N \sin{\left(\frac{n\pi x}{L}\right)[/tex]

where N is the yet unknown normalization constant. Then you plug this wavefunction into

[tex]1=\int_0^L \psi_n^*(x)\psi_n(x) dx[/tex]

and solve for N. Since both [itex]\psi_n^*(x)[/itex] and [itex]\psi_n(x)[/itex] contain N, you get [itex]N^2[/itex] in the equation.
 
  • #6
Ah right, it's been a while...I suppose N^2 would be appropriate. Otherwise, you'd square-root N. (I always use A instead of N which adds to my confusion...)
 
  • #7
Sorry about that, i just saw it in my textbook and copied it down, but Vela's explanation really cleared it up! thanks both of you guys. Sorry I am a newbie when it comes to this stuff... and I'm rusty on math.

Also would you guys know anything about guassian integrals? if i was trying to integrate e^(-2ax^2)?
 

1. What is the purpose of normalizing a wavefunction?

Normalizing a wavefunction is necessary to ensure that the total probability of finding a particle in any location is equal to 1. This is a fundamental requirement in quantum mechanics, as the square of the wavefunction gives the probability density of finding a particle in a particular location.

2. How is normalization achieved?

Normalization is achieved by dividing the wavefunction by a normalization constant. This constant is calculated by taking the square root of the integral of the square of the wavefunction over all space. This ensures that the total probability of finding the particle in any location is equal to 1.

3. Why is it important to normalize a wavefunction?

Normalizing a wavefunction ensures that the wavefunction is physically meaningful and accurately describes the system. If a wavefunction is not normalized, the probabilities calculated from it will not be accurate and may violate the fundamental principles of quantum mechanics.

4. Can any wavefunction be normalized?

No, not all wavefunctions can be normalized. A wavefunction must be square-integrable, meaning that the integral of the square of the wavefunction over all space must be finite, in order to be normalized. If the integral is infinite, the wavefunction cannot be normalized.

5. Does normalizing a wavefunction change its physical properties?

No, normalizing a wavefunction does not change its physical properties. The wavefunction may change in magnitude, but the overall shape and behavior of the wavefunction remain the same. Normalization only ensures that the wavefunction is a valid and accurate representation of the system.

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