Stationary states infinite cubic well

In summary, the stationary state of the infinite cubic well Hamiltonian is found to be e^{\frac{12 \hbar^2}{8mL^2}}.
  • #1
happyparticle
394
20
Homework Statement
For which values of c the state ##e^{c \cdot L_z} |2,2,2>## is stationary for the infinite cubic well Hamiltonian.
Relevant Equations
##\psi (r,t) = \sqrt{\frac{8}{l^3}} sin(\frac{2 \pi x}{l}) sin(\frac{2 \pi u}{l}) sin(\frac{2 \pi z}{l}) e^{-iEt/\hbar}##
##E = \frac{6}{m} (\frac{\pi \hbar}{l})^2##
For a state to be stationary it must be time independent.

Naively, I tried to find the values of c where I don't have any time dependency.

##e^{c \cdot L_z} \psi (r,t) = e^{c L_z} \sqrt{\frac{8}{l^3}} sin(\frac{2 \pi x}{l}) sin(\frac{2 \pi u}{l}) sin(\frac{2 \pi z}{l}) e^{-iEt/\hbar}##

##e^{c \cdot L_z -iEt/\hbar} ##

Thus
##c = \frac{L_z - iEt}{\hbar}##

##c = \frac{L_z - i\frac{6}{m} (\frac{\pi \hbar}{l})^2t}{\hbar}##

I'm wondering if this is correct.

Thanks
 
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  • #2
happyparticle said:
Homework Statement:: For which values of c the state ##e^{c \cdot L_z} |2,2,2>## is stationary for the infinite cubic well Hamiltonian.
Relevant Equations:: ##\psi (r,t) = \sqrt{\frac{8}{l^3}} sin(\frac{2 \pi x}{l}) sin(\frac{2 \pi u}{l}) sin(\frac{2 \pi z}{l}) e^{-iEt/\hbar}##
##E = \frac{6}{m} (\frac{\pi \hbar}{l})^2##

For a state to be stationary it must be time independent.

Naively, I tried to find the values of c where I don't have any time dependency.

##e^{c \cdot L_z} \psi (r,t) = e^{c L_z} \sqrt{\frac{8}{l^3}} sin(\frac{2 \pi x}{l}) sin(\frac{2 \pi u}{l}) sin(\frac{2 \pi z}{l}) e^{-iEt/\hbar}##

##e^{c \cdot L_z -iEt/\hbar} ##

Thus
##c = \frac{L_z - iEt}{\hbar}##

##c = \frac{L_z - i\frac{6}{m} (\frac{\pi \hbar}{l})^2t}{\hbar}##

I'm wondering if this is correct.

Thanks
A stationary state itself is not time independent. Its absolute square is time independent.
That won't get you very far though. You have to look at something else, I would say.
 
  • #3
The eigenfunction is ##
\sqrt{\frac{8}{l^3}} sin(\frac{2 \pi x}{l}) sin(\frac{2 \pi u}{l}) sin(\frac{2 \pi z}{l}) e^{-iEt/\hbar}
##
The eigenvalue is ##\frac{12 \hbar^2}{8mL^2}##
Maybe ##e^{c L_z} = e^{\frac{12 \hbar^2}{8mL^2}}##
 
  • #4
I assume ##L_z## is the operator corresponding to the z-component of orbital angular momentum, so your attempts don't make mathematical sense. You should go back and review the basics of operators and wave functions.

If it's not supposed to represent that operator, what is it supposed to be?
 

1. What is a stationary state in an infinite cubic well?

A stationary state in an infinite cubic well refers to a quantum state where the probability of finding a particle in a specific location does not change over time. This is because the potential energy within the well is constant, leading to a stable and unchanging state.

2. How is the energy of a particle in an infinite cubic well determined?

The energy of a particle in an infinite cubic well is determined by the quantum numbers nx, ny, and nz, which represent the number of nodes in the x, y, and z directions, respectively. The energy can be calculated using the equation E = (nx2 + ny2 + nz2)h2/8mL2, where h is Planck's constant, m is the mass of the particle, and L is the length of the well.

3. What is the significance of the infinite cubic well in quantum mechanics?

The infinite cubic well is a commonly used model in quantum mechanics to study the behavior of particles in confined spaces. It helps to understand the concept of energy quantization and the wave-like nature of particles, as well as to make predictions about their behavior in other potential energy systems.

4. Can a particle in an infinite cubic well have any energy value?

No, the energy of a particle in an infinite cubic well is quantized, meaning it can only have certain discrete values determined by the quantum numbers. The particle cannot have any arbitrary energy value within the well.

5. How does the size of the infinite cubic well affect the energy of a particle?

The energy of a particle in an infinite cubic well is inversely proportional to the size of the well. This means that as the well size increases, the energy levels become closer together, and the energy spacing approaches zero. As the well size decreases, the energy levels become more spread out, with larger energy spacings between them.

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