# Other ways of finding expectation value of momentum

• I
• VVS2000
In summary: I said I have already found the expectation value using ehrenfest theorem. just asked for some other ways that is all.
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
anuttarasammyak said:
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?

VVS2000 said:
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
anuttarasammyak said:
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

VVS2000 said:
I was solving a problem
What problem? Being more specific about the problem you are trying to solve might help.

topsquark
PeterDonis said:
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.

Last edited:
VVS2000 and topsquark
anuttarasammyak said:
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

## 1. How can I find the expectation value of momentum using the wave function?

The expectation value of momentum can be calculated by taking the integral of the product of the wave function and the momentum operator. This integral is known as the momentum expectation value integral.

## 2. Can I use the uncertainty principle to find the expectation value of momentum?

No, the uncertainty principle only gives a lower bound on the product of the uncertainty in position and momentum. It cannot be used to directly calculate the expectation value of momentum.

## 3. Are there any other mathematical methods for finding the expectation value of momentum?

Yes, the expectation value of momentum can also be found using the momentum probability distribution function. This function is derived from the wave function and can be used to calculate the expectation value.

## 4. Is the expectation value of momentum the same as the average momentum?

Yes, the expectation value of momentum is also known as the average momentum because it represents the average value of the momentum of a particle in a given state.

## 5. How does the expectation value of momentum relate to the physical motion of a particle?

The expectation value of momentum gives us information about the average momentum of a particle in a given state, but it does not tell us about the specific motion of a particle. Other physical quantities, such as the position and velocity, are needed to fully describe the motion of a particle.

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