# Other ways of finding expectation value of momentum

• I
• VVS2000
I said I have already found the expectation value using ehrenfest theorem. just asked for some other ways that is all.f

#### VVS2000

Apart from the usual integral method, are there any other ways to find expectation value of momentum? I know one way is by using ehrenfest theorem, relating it time derivative of expectation value of position operator.
Even using the uncertainty principle, we might get it if we know the uncertainty in position but not the exact value as we might get only the lower bound of it.
so, if any other ways, please do share!

The integral is expressed by momentum wave function ##\phi(p)## ,i.e.
$$\int \psi^{*}(x) \frac{\partial}{i \hbar \partial x} \psi(x) dx=\int \phi^{*} (p) p\ \phi(p) dp$$
where ##\psi(x)## and ##\phi(p)## are Fourier or inverse Fourier transform of the other.

• vanhees71 and VVS2000
The integral is expressed by momentum wave function ##\phi(p)## ,i.e.
$$\int \psi^{*}(x) \frac{\partial}{i \hbar \partial x} \psi(x) dx=\int \phi^{*} (p) p\ \phi(p) dp$$
where ##\psi(x)## and ##\phi(p)## are Fourier or inverse Fourier transform of the other.
yeah I mentioned the integral method in the OP
anything other than this?

anything other than this?
By symmetrical treatment of coordinate x and momentum p as is shown in Fourier transforms, your question applies similarly on coordinate x , i.e.," Are there ways to get expectation value <x> and <p> other than
$$<x>=\int \psi(x)^* x \psi(x) dx$$
$$<p>=\int \phi(p)^* p \phi(p) dp$$?"
I regard these formula as definitions of <x> and <p>. If you happen to find other ways, I am afraid you scarcely do, the values of which must coincide with the values given by them.

• VVS2000
By symmetrical treatment of coordinate x and momentum p as is shown in Fourier transforms, your question applies similarly on coordinate x , i.e.," Are there ways to get expectation value <x> and <p> other than
$$<x>=\int \psi(x)^* x \psi(x) dx$$
$$<p>=\int \phi(p)^* p \phi(p) dp$$?"
I regard these formula as definitions of <x> and <p>. If you happen to find other ways, I am afraid you scarcely do, the values of which must coincide with the values given by them.
Yeah I know there are few methods other than this
I only asked because I was solving a problem and solving the integral to find the expectation value was very tedious and long
Just wanted to know whether there are other ways of finding it, that is all

I was solving a problem
What problem? Being more specific about the problem you are trying to solve might help.

• topsquark
What problem? Being more specific about the problem you are trying to solve might help.
Expectation value of momentum of particle in a box and the wave function is time dependent as well

What is the initial condition of your case? As an easy case of symmetric momentum wave function
$$\phi(p,t)=\phi(-p,t)$$
Obviously <p>=0.

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• VVS2000 and topsquark
What is the initial condition of your case? As an easy case of symmetric momentum wave function
$$\phi(p,t)=\phi(-p,t)$$
Obviously <p>=0.
it is not symmetric, but as I said I have already found the expectation value using ehrenfest theorem. just asked for some other ways that is all