Expectation value of momentum times position particles in a box

In summary, the expectation value for <p*x> for a particle in a box is equal to -ih(bar)sin^2(nπx/l).
  • #1
adebola1
1
0

Homework Statement



What is the expectation value of <p*x> aka the momentum times the position operator, for a particle in a box.

Homework Equations



Psi(x) = root(2/l) sin (n∏x/l)
P= -ih(bar)d/dx
X=x

The Attempt at a Solution


All integrals are from 0 to L
I'm typing this on a playbook so I won't show all my steps.
-2ih/l ∫sin(n∏x/l)d/dx(xsin(nx/l)
Which I got to be : sinx(sinx) + xcosx. My teacher however never gave us the integral of xcosx from 0 to l. So how can I solve this?
 
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  • #2
adebola1 said:

Homework Statement



What is the expectation value of <p*x> aka the momentum times the position operator, for a particle in a box.

Homework Equations



Psi(x) = root(2/l) sin (n∏x/l)
P= -ih(bar)d/dx
X=x

The Attempt at a Solution


All integrals are from 0 to L
I'm typing this on a playbook so I won't show all my steps.
-2ih/l ∫sin(n∏x/l)d/dx(xsin(nx/l)
Which I got to be : sinx(sinx) + xcosx. My teacher however never gave us the integral of xcosx from 0 to l. So how can I solve this?

The integral of x*cos(x)dx is equal to the integral of x*d(sin(x)). Use integration by parts.
 

Related to Expectation value of momentum times position particles in a box

1. What is the expectation value of momentum times position for particles in a box?

The expectation value of momentum times position for particles in a box is a mathematical concept used to describe the average momentum and position of a particle in a confined space. It is calculated by taking the integral of the product of the momentum and position probability distributions over the volume of the box.

2. How is the expectation value of momentum times position related to Heisenberg's uncertainty principle?

The expectation value of momentum times position is related to Heisenberg's uncertainty principle, which states that it is impossible to know both the exact position and momentum of a particle at the same time. The uncertainty in the position and momentum of a particle is inversely proportional, meaning that as one becomes more certain, the other becomes less certain. The expectation value of momentum times position reflects this uncertainty by giving a measure of the average position and momentum of the particle.

3. Can the expectation value of momentum times position be negative?

Yes, the expectation value of momentum times position can be negative. This does not mean that the particle has a negative momentum or position, but rather that the average momentum and position of the particle are in opposite directions. This is possible because the expectation value takes into account the probability of the particle being in different positions and having different momenta.

4. How does the size of the box affect the expectation value of momentum times position?

The size of the box does not directly affect the expectation value of momentum times position. However, it can indirectly influence it by changing the probability distribution of the particle's position and momentum. A larger box will allow for more possible positions and momenta, resulting in a wider probability distribution and potentially affecting the expectation value.

5. How is the expectation value of momentum times position calculated?

The expectation value of momentum times position is calculated by taking the integral of the product of the momentum and position probability distributions over the volume of the box. This integral can be solved using mathematical techniques such as integration by parts or substitution. It is important to note that the calculation may differ depending on the specific system and potential energy function being studied.

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