Do All Lower Energy States Exist When n=3 in an Infinite Potential Well?

[victoria13]

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

 

[vela]

The particle is in a superposition of the n=3 state and the n=1 state.

 

[victoria13]

so the energies are when n=3 and n=1?

 

[vela]

Yes, the measurement can only result in energies of states which you have some chance of finding the particle in.

 

[paranormal]

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.