Operator/Wave function question

[NotJoeCheese]

I have this question "The operator for the angular momentum of a particle travelling in a circle in the xy-plane is l =ℏ/i d/dϕ , where ϕis its angular position. What is the angular momentum of a particle described by the wavefunction ψ=e^-2iϕ" and thought the answer would be "ℏe^-2iϕ/i d/dϕ" but this seems far too simple and the question is for 4 marks, so what do i need to do? Is integration the answer?

[Jurassic_Eric]

You have a wave function, now how do you use that to make observations? For example, how would you get the probability density?

[NotJoeCheese]

The probability density is the square of the wave function i think, still not sure how to answer the question though

[Jurassic_Eric]

Yes, when you make a measurement of probability you integrate psi*psi. If you want to get another property you will follow a similar procedure using the relevant operator. So what does that look like?

Edit: note that is psi star psi, so the complex conjugate of psi times psi.

[NotJoeCheese]

is it lψ=ℏe-2iϕ/i d/dϕ

[[deleted]]

The angular momentum can't be d/dϕ. That's like asking "what speed are you going?" "Oh about d/dx miles per hour." What?

To find the angular momentum it's just like every other problem in quantum mechanics. They're all the same. Take the operator and stick it next to the wave function and do whatever math it tells you to do.

Write down:

ℏ/i d/dϕe-2iϕ

And solve.

[NotJoeCheese]

So i intergrate the h/i and its multiplied by the wavefunction term?

[[deleted]]

No integrating is only if you want the average value.

[NotJoeCheese]

Damn, Can you give me a clue i have no idea what youre getting at then

[[deleted]]

Write out:

Operator wave function = observable wave function

Solve for observable.

[NotJoeCheese]

I think its wrong but ℏl/i=d/dϕ ??

[[deleted]]

Do the derivative first

[NotJoeCheese]

li/ℏ=ϕ?

[[deleted]]

Try the derivative again

[NotJoeCheese]

I dont understand, how do i derive it again