Infinite Square Well - Particle in linear combination of states

In summary: The clever answer is that this problem is ill-posed, because strictly speaking a particle in an infinite-square well has no momentum observable, but I'm pretty sure that this is not what the professor wants to know ;-).So guess you are supposed to find c_0 and c_1 such that the naive "momentum"-expectation value\int_{-L/2}^{L/2} \mathrm{d} x \; \psi^*(x,t) \left (-\hbar \frac{\mathrm{d}}{\mathrm{d} x} \psi(x,t) \right )=\text{max},
  • #1
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A particle of mass m is trapped in a one-dimensional infinite square well running from x= -L/2 to L/2. The particle is in a linear combination of its ground state and first excited state such that its expectation value of momentum takes on its largest possible value at t=0.I know the process of solving PDE's clear as day, that's not the issue. The problem is that I'm tripping myself out on how to write [itex]\Psi(x,t)[/itex] as a linear combination of its ground state + first excited state.

My hunch is to approach the problem like this :

[itex]\Psi(x,t)[/itex]=c[itex]_{0}[/itex][itex]\Psi_{0}(x,t)[/itex]+c[itex]_{1}[/itex][itex]\Psi_{1}(x,t)[/itex]

where 0 and 1 represent the ground state and first state, respectively

I'm confusing myself on what needs to fill in the equations! Any help would be appreciated.
 
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  • #2
The clever answer is that this problem is ill-posed, because strictly speaking a particle in an infinite-square well has no momentum observable, but I'm pretty sure that this is not what the professor wants to know ;-).

So guess you are supposed to find [itex]c_0[/itex] and [itex]c_1[/itex] such that the naive "momentum"-expectation value
[tex]\int_{-L/2}^{L/2} \mathrm{d} x \; \psi^*(x,t) \left (-\hbar \frac{\mathrm{d}}{\mathrm{d} x} \psi(x,t) \right )=\text{max},[/tex]
where [itex]\psi[/itex] must be properly normalized, i.e.,
[tex]\int_{-L/2}^{L/2} \mathrm{d} x \; |\psi(x,t)|^2=0.[/tex]
 
  • #3
vanhees71 said:
The clever answer is that this problem is ill-posed, because strictly speaking a particle in an infinite-square well has no momentum observable, but I'm pretty sure that this is not what the professor wants to know ;-).

So guess you are supposed to find [itex]c_0[/itex] and [itex]c_1[/itex] such that the naive "momentum"-expectation value
[tex]\int_{-L/2}^{L/2} \mathrm{d} x \; \psi^*(x,t) \left (-\hbar \frac{\mathrm{d}}{\mathrm{d} x} \psi(x,t) \right )=\text{max},[/tex]
where [itex]\psi[/itex] must be properly normalized, i.e.,
[tex]\int_{-L/2}^{L/2} \mathrm{d} x \; |\psi(x,t)|^2=0.[/tex]

thanks! that makes sense. By the way, the problem wants me to find the nomalized ψ(x,t) and the usual dynamical quantities (<x>, <p>, check m(d/dx)<x>=p, Δx, Δp, check ΔxΔp satisfies uncertainty principle)

how do we write ψ(x,t) as a linear combination of its ground and first excited state like the problem asks??
 
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  • #4
Just like you did in the original post.
 
  • #5
I'm stuck on the fact that were given the phrase "expectation value of the momentum is a maximum when t=0."

the energy of the second state is higher than that of the first, which implies the momentum of the second state which is more than that in the first (ps- I'm just talking about expectation values)

Does anyone know what this means ?
 
  • #6
Still confused about that phrase, if anyone has ideas please let me know ! Thanks
 

1. What is the Infinite Square Well model?

The Infinite Square Well model is a simplified system used in quantum mechanics to describe the behavior of a particle confined to a one-dimensional space. It consists of a potential well with infinitely high walls, representing an infinite potential barrier that the particle cannot penetrate.

2. What is the equation for the wave function of a particle in the Infinite Square Well?

The equation for the wave function of a particle in the Infinite Square Well is ψ(x) = A sin(nπx/L), where A is a normalization constant, n is the quantum number, and L is the width of the well.

3. How does the energy of a particle in the Infinite Square Well depend on its quantum number?

The energy of a particle in the Infinite Square Well is directly proportional to its quantum number, with the lowest energy state having a quantum number of n=1 and the energy increasing as n increases.

4. Can a particle in the Infinite Square Well occupy multiple states at the same time?

Yes, a particle in the Infinite Square Well can exist in a linear combination of states, meaning it can simultaneously occupy multiple energy states with different probabilities. This is known as superposition and is a fundamental concept in quantum mechanics.

5. How does the probability of finding a particle in the Infinite Square Well change over time?

The probability of finding a particle in the Infinite Square Well oscillates over time, with the highest probability occurring at the center of the well and decreasing towards the walls. This oscillation is known as the particle's wave function collapse and is a result of the particle's energy states interfering with each other.

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