Particle in a box momentum uncertainty

In summary, this problem asks you to find the cn's for the new eigenstates of the Hamiltonian, and then find the probability of the particle being in a particular state.
  • #1
Vaal
42
0
1.
this is for a particle in a box (infinite potential well):
For the wave functions calculate (as a function of n)
[tex]\Delta[/tex]x
[tex]\Delta[/tex]p
[tex]\Delta[/tex]x[tex]\Delta[/tex]p

Homework Equations


h = h bar below

u+n(x)=(1/[tex]\sqrt{a}[/tex] )cos((n-1/2)(pi)x/a)
u-n(x)=(1/sqrt{a})sin(n(pi)x/a)
E-n=h2/(2m)(n(pi)/a)2
E+n=h2/(2m)((n-1/2)(pi)/a)2

The Attempt at a Solution


ok i think the Delta x is easy, just find the variance of u+n(x)2 and u-n(x)2 as a function of n (correct me if i am wrong)

where i am lost is on the momentum, i can obviously get it from the energy but i don't see how to determine its variance or even how it would change within one state. i would just use the uncertainty principle but that seems to be what they want you to show in the next step (also it is an inequality not an exact equation)

thanks for any help and sorry if this is an absurd question, it has been awhile since i have done any of this sort of stuff.
 
Physics news on Phys.org
  • #2
Find [itex]\langle \hat{p}^2 \rangle[/itex] where

[tex]\hat{p} = \frac{\hbar}{i}\frac{\partial}{\partial x}[/tex]
 
  • #3
Ok, I think I figured out the first part described above, although I can't seem to make the numbers come out right.

I am lost on the next problem, even more so than the last.

My well currently has walls at +/- a and a particle is in the ground state. the walls are suddenly moved to +/- b with b>a,
what is the probability the particle will be in ground state of new potential? & what is probability of the particle being in first excited state?

i realize the ground state of the new potential has less energy so the particle is more likely to be at a higher state than it was in the old potential but i have no idea how to make it quantitive and get the exact probabilities. any help would be greatly appreciated. thanks
 
  • #4
Vaal said:
My well currently has walls at +/- a and a particle is in the ground state. the walls are suddenly moved to +/- b with b>a,
what is the probability the particle will be in ground state of new potential? & what is probability of the particle being in first excited state?

i realize the ground state of the new potential has less energy so the particle is more likely to be at a higher state than it was in the old potential but i have no idea how to make it quantitive and get the exact probabilities. any help would be greatly appreciated. thanks
Let the particle be in the state [itex]|\psi\rangle[/itex] and let [itex]|\phi_n\rangle[/itex] for n=1, 2, 3, … denote the eigenstates of the new Hamiltonian. You can express [itex]|\psi\rangle[/itex] in terms of the new eigenstates:

[tex]|\psi\rangle = c_0|\phi_0\rangle + c_1|\phi_1\rangle + \cdots[/tex]

where the cn's are constants, possibly complex. You need to find what the cn's are for n=0 and n=1. The probability of finding the particle in a particular state is equal to the modulus squared of the corresponding constant.
 
  • #5


Your approach to finding the uncertainty in position is correct. To find the uncertainty in momentum, you can use the relation \Deltap = h/\Deltax. This means that the uncertainty in momentum is inversely proportional to the uncertainty in position. So as the uncertainty in position decreases, the uncertainty in momentum increases and vice versa. This is in line with the uncertainty principle, which states that it is impossible to know the exact position and momentum of a particle simultaneously.

To find the uncertainty in momentum as a function of n, you can use the equations for the energy levels E+n and E-n. Then, use the relation E = p^2/2m to find the momentum and then plug it into the uncertainty equation. You should find that the uncertainty in momentum increases as n increases, since the energy levels and therefore the momentum also increase with n.

Finally, to find the product of the uncertainties, you can simply multiply the uncertainties in position and momentum. Since they are inversely proportional, the product should be constant for all values of n. This shows the validity of the uncertainty principle for a particle in a box.
 

What is a particle in a box?

A particle in a box refers to a theoretical model in quantum mechanics where a particle is confined to a one-dimensional box with reflective walls. This model allows for the study of the behavior and properties of the particle within the box.

What is momentum uncertainty in the context of a particle in a box?

Momentum uncertainty in the context of a particle in a box refers to the uncertainty or variation in the momentum of the particle. In quantum mechanics, the position and momentum of a particle cannot be known simultaneously with absolute certainty, and the Heisenberg uncertainty principle states that there will always be a degree of uncertainty in these values.

How is momentum uncertainty related to the size of the box?

The size of the box has a direct effect on the momentum uncertainty of the particle. As the size of the box decreases, the available space for the particle to move in decreases as well, leading to a larger momentum uncertainty. Conversely, a larger box allows for more possible momentum values, resulting in a smaller momentum uncertainty.

What is the formula for calculating momentum uncertainty in a particle in a box?

The formula for calculating momentum uncertainty in a particle in a box is given by: Δp = (h/2L), where Δp is the uncertainty in momentum, h is the Planck's constant, and L is the length of the box. This formula is derived from the Heisenberg uncertainty principle.

Why is momentum uncertainty important in quantum mechanics?

Momentum uncertainty is important in quantum mechanics because it is a fundamental concept that reflects the probabilistic nature of particles at the quantum level. It also plays a crucial role in understanding and predicting the behavior and properties of particles in various systems, including the particle in a box model.

Similar threads

  • Advanced Physics Homework Help
Replies
15
Views
2K
  • Advanced Physics Homework Help
Replies
24
Views
680
  • Advanced Physics Homework Help
Replies
9
Views
1K
  • Advanced Physics Homework Help
Replies
24
Views
2K
Replies
3
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
3
Views
1K
  • Advanced Physics Homework Help
Replies
3
Views
942
  • Advanced Physics Homework Help
Replies
7
Views
1K
Back
Top