Particle in a Box: Energy Measurements and Probabilities Explained

In summary, the conversation discusses a particle in an infinite square well with a state described as a linear combination of the two lowest energy states. The question asks about the possible results of a measurement of energy, the associated probabilities, and the average value of energy. The responses suggest using the quantum mechanical rules and working out the equations for two particle systems.
  • #1
phrygian
80
0

Homework Statement



A particle is in an infinite square well extending from x = 0 to x = a. Its state is a
linear combination of the two lowest energy states.
Psi(x, t) = A(2 Psi1(x, t) + Psi2(x, t))
a) If a measurement of energy is made what are the possible results of the measurement?
What is the probability associated with each? What is the average value of energy?


Homework Equations





The Attempt at a Solution



I know that when the Hamiltonian operates on Psi_n(x,t) it give E_n Psi_n(x,t), right? So are the possible results just Psi1(x,t) and Psi2(x,t) or would they be 2A E1 Psi1(x,t) and A E2 Psi2(x,t)? I am confused because I know quantum state vectors are supposed to be normalized but I am confused how that translates to wave functions?

And as for the probability, I really am not sure where to start can someone point me in the right direction?

Thanks for the help
 
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  • #2
The possible outcomes of a measurement are the eigenvalues of the corresponding observable. The coefficients of the wavefunction, written as a superposition of eigenstates, will tell you what those respective probabilities are.

This question is really just a straightforward application of the quantum mechanical rules (postulates) so it's hard to help without giving the answer.
 
  • #3
phrygian said:

Homework Statement



A particle is in an infinite square well extending from x = 0 to x = a. Its state is a
linear combination of the two lowest energy states.
Psi(x, t) = A(2 Psi1(x, t) + Psi2(x, t))
a) If a measurement of energy is made what are the possible results of the measurement?
What is the probability associated with each? What is the average value of energy?


Homework Equations





The Attempt at a Solution



I know that when the Hamiltonian operates on Psi_n(x,t) it give E_n Psi_n(x,t), right? So are the possible results just Psi1(x,t) and Psi2(x,t) or would they be 2A E1 Psi1(x,t) and A E2 Psi2(x,t)? I am confused because I know quantum state vectors are supposed to be normalized but I am confused how that translates to wave functions?

And as for the probability, I really am not sure where to start can someone point me in the right direction?

Thanks for the help


First of all, all this equation states: ''Psi(x, t) = A(2 Psi1(x, t) + Psi2(x, t))'' is two particles oscillating through a specific position and a time. You must state they are in a superpositioning, which is actually a type of interference. The total energy of one of these particles is:

[itex]E=\hbar \omega[/itex]. It also has a respective wave number [itex]k[/itex]: [itex]\frac{\hbar^{2}k^{2}}{2m}[/itex]. Now work it out for two particle systems.
 
  • #4
ManyNames said:
First of all, all this equation states: ''Psi(x, t) = A(2 Psi1(x, t) + Psi2(x, t))'' is two particles oscillating through a specific position and a time. You must state they are in a superpositioning, which is actually a type of interference. The total energy of one of these particles is:

[itex]E=\hbar \omega[/itex]. It also has a respective wave number [itex]k[/itex]: [itex]\frac{\hbar^{2}k^{2}}{2m}[/itex]. Now work it out for two particle systems.
The problem statement clearly says the state describes one particle, not two.
 
  • #5
vela said:
The problem statement clearly says the state describes one particle, not two.

Then the equation required, the one given is not the ordinary standard equation i have ever seen. This is the one which will calculate the question at hand.

[itex]\psi(x,t) = [A sin(k,t)+B cos(k,t)]e^{-i \omega t}[/itex]
 
  • #6
ManyNames said:
Then the equation [...] given is not the ordinary standard equation i have ever seen.
Really? This is a standard question in introductory quantum mechanics.
 
  • #7
vela said:
Really? This is a standard question in introductory quantum mechanics.

Then why is there a factor of 2 for both psi identities which appear inside the brackets?
 
  • #8
What's a psi identity?

[itex]\Psi(x,t)[/itex] isn't an energy eigenstate. It's a superposition of the energy eigenstates [itex]\psi_1(x,t)[/itex] and [itex]\psi_2(x,t)[/itex].
 

What is a "Particle in a Box" in the context of energy measurements?

A "Particle in a Box" is a theoretical model used in quantum mechanics to explain the behavior of a particle confined within a specific region. In this model, the particle is assumed to have a discrete set of energy levels, similar to the rungs of a ladder, and can only exist within the boundaries of the box.

How are energy measurements obtained for a "Particle in a Box"?

Energy measurements for a "Particle in a Box" are obtained by solving the Schrödinger equation, which describes the behavior of quantum particles. This equation takes into account the size of the box, the mass of the particle, and the potential energy within the box. The solutions to this equation yield the energy levels and corresponding probabilities for the particle.

What is the significance of energy levels in a "Particle in a Box"?

The energy levels in a "Particle in a Box" represent the allowed states that the particle can have within the box. Each energy level has a corresponding probability, which describes the likelihood of finding the particle in that particular state. These energy levels and probabilities are crucial in understanding the behavior of quantum particles and their interactions.

What is the relationship between energy and probability in a "Particle in a Box"?

In a "Particle in a Box" system, the energy and probability are directly related. This means that the higher the energy level, the higher the probability of finding the particle in that state. Conversely, the lower the energy level, the lower the probability of finding the particle in that state. This relationship is described by the Schrödinger equation and plays a crucial role in understanding the behavior of quantum particles.

How does the size of the box affect the energy levels and probabilities in a "Particle in a Box" system?

The size of the box has a direct influence on the energy levels and probabilities in a "Particle in a Box" system. A larger box will result in a larger range of energy levels and a higher probability for the particle to occupy higher energy states. Conversely, a smaller box will have a smaller range of energy levels and a higher probability for the particle to occupy lower energy states. This relationship is described by the Schrödinger equation and is an essential factor in understanding the behavior of quantum particles.

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