A particle in an infinite square well

In summary, the conversation discusses the concepts of probability density and stationary state in relation to quantum states. The probability density is likely referring to the probability density function, while a stationary state means that the quantum state does not change over time. The problem involves finding the probability density of a given quantum state, which can be interpreted by looking it up in a textbook or online."
  • #1
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Homework Statement
The homework is in a photo, a long with some to most of my work.
Relevant Equations
There are a lot of equations....
Screenshot 2023-03-12 at 10.33.28 PM.png

What I am lost about is b, rather the rest of B. I am not sure what it means by probability density and a stationary state.
Screenshot 2023-03-12 at 10.03.01 PM.png
 
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  • #2
By probability density, it most likely means the probability density function. When a quantum state is stationary (and remember that it's the hamiltonian's eigenvector), that just means it does not grow throughout the course of time. That's how i'd put it.
 
  • #3
You have written down ##\psi(x,t)##. The problem is asking you to find "the probability density ##|\psi(x,t)|^2.## How do you interpret that?
Your textbook must have a definition for stationary state. If you cannot find it, look it up on the web.
 
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1. What is a particle in an infinite square well?

A particle in an infinite square well is a theoretical model used in quantum mechanics to study the behavior of a particle confined to a one-dimensional box with infinitely high potential walls on either side.

2. What are the properties of a particle in an infinite square well?

The properties of a particle in an infinite square well include a discrete energy spectrum, meaning the particle can only have certain energy levels, and a probability distribution that is confined to the well and decreases towards the walls.

3. How does the energy of a particle in an infinite square well relate to its wavelength?

The energy of a particle in an infinite square well is directly proportional to its wavelength. This is known as the de Broglie relation and is given by the equation E = (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 well.

4. What is the significance of the wave function in a particle in an infinite square well?

The wave function in a particle in an infinite square well represents the probability amplitude of finding the particle at a specific position within the well. It also describes the energy states and behavior of the particle within the well.

5. How does the behavior of a particle in an infinite square well differ from that of a free particle?

A particle in an infinite square well is confined to a specific region and has a discrete energy spectrum, while a free particle can move freely and has a continuous energy spectrum. Additionally, the wave function for a particle in an infinite square well is zero at the walls, while a free particle's wave function extends to infinity.

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