Infinite potential well

In summary, the conversation discusses a question about infinite potential wells and the possible energy states of a particle trapped within them. Through calculations and the use of a given hint, it is determined that the particle is in a superposition of the n=3 state and the n=1 state, meaning that the possible energies are when n=3 and n=1.
  • #1
victoria13
7
0
ok so i have a question on infinite potential wells... if you have the energy state when n=3, does that mean that you have to have the energy state when n=2 and n=1?

Ok basically the Q we are given is:

A particle of mass m is trapped in an infi nitely deep one-dimensional potential well between x = 0 and x = a, and at a time t = 0 is described by the wave function

psi(x,t=0) = sin(pix/a)cos(2pix/a)

(i) What possible values may be found for the energy of the particle?
(ii) What is the expectation (or average) value of the energy of the particle?
(iii) Give an expression showing the time dependence of the wave function, psi(x,t).
[Hint: Use sinAcosB = 1/2(sin(A + B) + sin(A-B))].

so when i use the hint and find the equation in terms of just sin i get

psi(x) = 0.5sin(3pix/a) + 0.5sin(-pix/a)

so does this mean that there is just the 3rd energy state, or the third and first, or all 3?

thanks
 
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  • #2
The particle is in a superposition of the n=3 state and the n=1 state.
 
  • #3
so the energies are when n=3 and n=1?
 
  • #4
Yes, the measurement can only result in energies of states which you have some chance of finding the particle in.
 
  • #5


I would like to clarify that the concept of an infinite potential well is a simplified model used in quantum mechanics to study the behavior of particles in confined spaces. In this model, the particle is assumed to be confined within a potential well with infinite walls, meaning it cannot escape.

To answer your question, the possible energy states for a particle in an infinite potential well are given by the equation E_n = (n^2*h^2)/(8mL^2), where n is the quantum number, h is Planck's constant, m is the mass of the particle, and L is the length of the potential well. This means that the energy states are discrete and depend on the value of n.

In your specific example, the particle is described by the wave function psi(x,t=0) = sin(pix/a)cos(2pix/a), where a is the length of the potential well. This wave function can be expanded using the hint given, leading to the expression psi(x) = 0.5sin(3pix/a) + 0.5sin(-pix/a). This means that the particle can have a superposition of the third and first energy states, and therefore, also the second energy state.

To find the expectation value of the energy, we use the equation <E> = ∫ psi*(x) * H * psi(x) dx, where H is the Hamiltonian operator. In this case, H = -(h^2/2m) * d^2/dx^2. By solving the integral, we can find the average energy of the particle.

In conclusion, the particle in an infinite potential well can have multiple energy states, and the specific energy state it occupies depends on the wave function that describes it. The concept of superposition allows for the particle to exist in multiple energy states simultaneously. I hope this helps clarify your question.
 

1. What is an infinite potential well?

An infinite potential well is a theoretical model used in quantum mechanics to study the behavior of a particle confined to a specific region with infinite potential energy barriers on either side.

2. How is the infinite potential well used in quantum mechanics?

The infinite potential well is used as a simplification of more complex potential energy functions to analyze the energy states and behavior of particles in confinement.

3. What are the energy levels in an infinite potential well?

In an infinite potential well, the energy levels are discrete and quantized, meaning they can only take on certain values determined by the size of the well.

4. What happens to the energy levels as the size of the infinite potential well changes?

As the size of the infinite potential well increases, the energy levels become closer together, and as the size decreases, the energy levels become more spread out.

5. Can a particle in an infinite potential well escape its confinement?

No, a particle in an infinite potential well cannot escape its confinement as the infinite potential energy barriers on either side prevent it from having enough energy to overcome them.

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