Time evolution of a particle in an ISW after the well

In summary, the state of a particle in an infinite square well will remain undisturbed if the width of the well is doubled and the particle starts in the ground state of the original well. This is because the initial wave function can be expressed as the product of the original wave function and the evolution operator, which is dependent on the particle's energy and time. The new energy eigenstate wavefunctions for the expanded well will determine the particle's time evolution in the larger well.
  • #1
Ratpigeon
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Homework Statement



How does the state of a particle in an ISW evolve with time after the width of the well doubles - from a to 2a
If the particle starts in the ground state of the half width well, then immediately after the well doubles it will be undisturbed therefore the initial wave function is
|P(0)>=Integrate[Sqrt[2/a] Sin[3.1415... x/a], {x,0,a}] |x>
THe evolution operator is
|P(t)>=Exp[-i E t/hbar] |P(0)>

Homework Equations


The Attempt at a Solution


Does this mean that the solution is just stitching those two equations together? I plotted it and it came out that the solution just went up and down in the half well (or, if I changed the integration limits, it came out at just the second excited state of the full well).
I was under the impression that some sort of spreading had to occur, but I'm not sure how to get it...
 
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  • #2
What are the new energy eigenstate wavefunctions for the expanded well? These become the states that have a simple time evolution.

[EDIT: I don't understand this expression: |P(0)>=Integrate[Sqrt[2/a] Sin[3.1415... x/a], {x,0,a}] |x>]
 
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FAQ: Time evolution of a particle in an ISW after the well

What is the ISW (Infinite Square Well) and how does it affect the time evolution of a particle?

The ISW is a theoretical model used in quantum mechanics to represent a particle confined within a finite region of space with infinite potential barriers. The potential energy within the well is constant, but outside the well, it is infinitely high. This confinement affects the time evolution of a particle by limiting its possible energy states and causing it to behave differently from a free particle.

How does the time evolution of a particle change after it enters the well?

After entering the ISW, the particle's wave function will undergo a series of oscillations as it bounces back and forth between the potential barriers. The amplitude of these oscillations decreases over time as the particle's energy is gradually dissipated through the barriers. The particle's position and momentum will also become more uncertain as time progresses.

What is the role of the Schrödinger equation in determining the time evolution of a particle in an ISW?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes how a particle's wave function evolves over time. By solving this equation for the specific conditions of the ISW, we can predict the time evolution of the particle's wave function and its associated properties, such as position and momentum.

Can the time evolution of a particle in an ISW be observed experimentally?

While we cannot directly observe the wave function of a particle, we can indirectly measure its properties, such as position and momentum, through various experimental techniques. By performing repeated measurements on a large number of particles in an ISW, we can confirm the predictions made by the Schrödinger equation and validate our understanding of the time evolution of these particles.

How does the time evolution of a particle in an ISW differ from that of a free particle?

A free particle can have any energy state and can move freely through space, whereas a particle in an ISW is confined to a specific region and has a limited number of energy states. As a result, the time evolution of a free particle is continuous and smooth, while that of a particle in an ISW is characterized by discrete energy states and oscillations between them. Additionally, the uncertainty in the particle's position and momentum is greater in an ISW due to the confinement and potential barriers.

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