QM. subjective questions about wavefunction

[NepToon]

Homework Statement

Nothing that big, just some questinos that i had about wavefunctions. i was reading this handout and came across this.
Suppose i am given an equation of a wave function, how do i know whether or not does it describe the state of definite energy and/or in the state od definite momentum.

Homework Equations

i just took an equation from the handout
ѱ=Asin²(nx)

The Attempt at a Solution

I normalized the function, and then I didnt really have anywhere to go.

 

[malawi_glenn]

This is a quite good tutorial on that subject I think

http://physics.nmt.edu/~raymond/classes/ph13xbook/node94.html

 

[NepToon]

hey, its you again.
so i did some reading and about eigenfunction, how do i find the eigenfunction of a wavefunction? like the one i mentioned earlier? does the state of definite energy mean whether or not is the energy an eigenfunction of the given function?

 

[malawi_glenn]

careful of the language.

You don't find the eigenfunction of a wavefunction, you determine IF a given wavefunction IS an eigenfuction to a given operator.

now, turning to your ѱ=Asin^2(nx), you can do the same arguments as in that web-page

 

[NepToon]

"you determine IF a given wavefunction IS an eigenfuction to a given operator"
so i find the eigenfunction of the operator and see if it is any similar to the wavefunction?

 

[malawi_glenn]

no, you operate with the operator on the given wavefunction and se if you get the same thing back. For instance psi(x) = x^2 is not an eigenfunction to the momentum operator.

 

[NepToon]

okay so i operate the momentum operator on ѱ=Asin²(nx) ie find p(with hat) ѱ? I've been scanning through this forum and see "psi" a lot, what does psi mean?

 

[malawi_glenn]

psi =\Psi , \qquad \psi

A greek letter, very often used to label the wavefunction.

Now. plane wave solutions are a bit trickier, sin^2(nx) is not an eigenfunction to the momentum operator, but it carries momentum n*hbar.

What book do you use in your course?

 

[NepToon]

we don't have a book. my teacher provides us with handouts, daily. why is ѱ=sin² nx not an eigenfunction of the momentum operator?
how does it carry the ih bar; i was just wondering as it is not in the equation of the wavefunction.

 

[malawi_glenn]

well, hmm the derivative of sin^2 x with respect to x is 2sinx cosx... so it is not an eigenfunction.

The hbar is there for uniti considerations. n is the wavenumber, and has units length^-1, so one has to have hbar to make units of momentum (mass*velocity)

Now sin^2 x is a WAVEfunction in that sense that it describes a wave, but not a QUANTUM wave. Many textbooks starts to introduce and recall properties of waves before moving on to quantum waves and the Schrödinger equation. That is why I asked which book you used.

Here are some good lecture notes

https://www.physicsforums.com/showthread.php?t=220904

https://www.physicsforums.com/showthread.php?t=220901

 

[NepToon]

so does that mean trig functions like
ѱ=sin (kx) cos(kx) or other such functions are not in the state of an energy eigenfunction?
if they are not an energy eigenfunction, is it possible for them to be a momentum eigenfunction, or not?

 

[malawi_glenn]

We have only discussed momentum so far.

Energy is (h-bar^2 momentum^2) /(2*Mass)

 

[NepToon]

well since energy=(momentum)² /2*mass, and they are related; if it is not an eigen function of energy operator, is it possible for it to be a eigenfunction of the momentum operator then?

 

[malawi_glenn]

what is the second derivative on sine-function? Recall our earlier discussion today! ;-)

sin^2(x) can be eigenfunction to p^2, but not p. You have to try.

Sin(x) is not eigenfunction to p, but it is an eigenfunction for p^2.

You can't argue the way you just did.

 

[NepToon]

so trig functions need not be the eigen function of energy operator but they can be eigenfunction of momentum operator??
but from the last post it sounds like the other way around.

 

[malawi_glenn]

No, you are mixing things up.

You argued since that sin ^2 x is not eigenfunction of p, then it can't be eigenfunction to p^2. But I just demonstrated that you can have an eigenfunction to p^2, even if that function is not an eigenfunction to p.

 

[NepToon]

so the wave function
ѱ=Asin²(nx) is an eigenfunction of momentum(why?) but not energy,
while
ѱ=sin (kx) cos(kx) is not an eigen function of energy or momentum?

 

[malawi_glenn]

Look at post 8,9 & 10

 

[NepToon]

okay this is getting way to confusing; can you just tell me, what conditions are needed for any given wavefunction to be the eigenfunction of:
a. energy operator
b. momentum operator
c. position operator

 

[malawi_glenn]

Why is it confusing? I have been totally coherent.

A function F is an Eigenfunction to an operator O, if:

O F= o*F

o is then said to be the Eigenvalue to the operator O on the eigenfunction F.

For instance, the function G is an eigenfunction to the derivative operator D, if DG=d*G, where d is some number, called the Eigenvalue of the derivative operator on the eigenfunction G.

 

[NepToon]

ok. its just the amount of post, that is making it confusing. u have been more than coherent. thnks for the help; epsecially the exercises, lecture notes and other references, really helps. i will go work on it and get some sleep. thnks for the time and effort glenn.

 

[malawi_glenn]

An advice is that after you have got some help, read the answer a few times, do a try on your own, if it does not work, try again, if it does not work then either - consult a book or webpage on the same matter. If that does not work, ask here again. That is the way you would ask for help from your teacher.

The 'problem' with quantum physics is that it is not as intuitive as classical physics, one has to practice the formalism and be very axiomatic - just as in a math class.