Quantum Wave Function: Infinite Sheet of Charge & Pinhole

In summary, the conversation discusses the formula for Psi(x) in the context of an infinite, positive, uniform sheet of charge with a pinhole in it. The electric field outside the sheet is constant and follows Gauss Law, leading to a "vee" shaped potential well. The solution to Schrodinger's equation involves matching decaying exponentials outside the well, which can be tedious without the use of a computer. The speaker also mentions being 72 years old and not finding this particular potential in their limited supply of quantum mechanics texts.
  • #1
GRDixon
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0
Imagine an infinite, positive, uniform sheet of charge with a pinhole in it. A negative particle oscillates back and forth through the pinhole and in the +-x direction. The magnitude of the force on it is constant in time (although the force reverses direction when the particle passes through the pinhole). Can anyone tell me what the formula for Psi(x) would be? Thanks. PS, I'm 72 years old, and this is not a homework problem. I just haven't found this particular potential in any of my limited supply of quantum mechanics texts.
 
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  • #2
GRDixon said:
Imagine an infinite, positive, uniform sheet of charge with a pinhole in it. A negative particle oscillates back and forth through the pinhole and in the +-x direction. The magnitude of the force on it is constant in time (although the force reverses direction when the particle passes through the pinhole). Can anyone tell me what the formula for Psi(x) would be? Thanks. PS, I'm 72 years old, and this is not a homework problem. I just haven't found this particular potential in any of my limited supply of quantum mechanics texts.

I assume you want to find the energy eigenfunctions in the position representation, which means we must solve the energy eigenequation, aka Schrodinger's time independent equation [tex] - \frac{{\hbar ^2 }}{{2m}}\frac{{d^2 \psi (x)}}{{dx^2 }} + V(x)\psi (x) = E\psi (x)[/tex]. From Gauss Law, we know that the electric field outside of an infinite charged sheet is constant, so that [tex]V(x) = ax[/tex]. This problem then is equivalent to a particle in a "vee" shaped potential well. This solution to Schrodinger's equation is in terms of Airy functions. The two constants of intergration are then obtained by matching [tex]\psi (x)[/tex] with the two decaying exponentials outside the well. Very tedious, unless you use a computer.
Best wishes
 
  • #3
eaglelake said:
I assume you want to find the energy eigenfunctions in the position representation, which means we must solve the energy eigenequation, aka Schrodinger's time independent equation [tex] - \frac{{\hbar ^2 }}{{2m}}\frac{{d^2 \psi (x)}}{{dx^2 }} + V(x)\psi (x) = E\psi (x)[/tex]. From Gauss Law, we know that the electric field outside of an infinite charged sheet is constant, so that [tex]V(x) = ax[/tex]. This problem then is equivalent to a particle in a "vee" shaped potential well. This solution to Schrodinger's equation is in terms of Airy functions. The two constants of intergration are then obtained by matching [tex]\psi (x)[/tex] with the two decaying exponentials outside the well. Very tedious, unless you use a computer.
Best wishes

Many Thanks. GRD
 

1. What is the definition of a quantum wave function?

A quantum wave function is a mathematical representation of the state of a quantum system, which describes the probability of finding a particle at a certain location or with a certain energy.

2. How does an infinite sheet of charge affect the quantum wave function?

An infinite sheet of charge can cause the quantum wave function to have a sharp transition at the location of the sheet, known as a discontinuity. This is because the electric field created by the sheet forces the wave function to change abruptly.

3. What happens to the quantum wave function when it passes through a pinhole?

When a quantum wave function passes through a pinhole, it will diffract, or spread out, due to the wave-like nature of particles. This results in a pattern of interference and diffraction on a detection screen beyond the pinhole.

4. How does the size of the pinhole affect the quantum wave function?

The size of the pinhole affects the diffraction pattern of the quantum wave function. A smaller pinhole will result in a wider diffraction pattern, while a larger pinhole will result in a narrower diffraction pattern.

5. Can the quantum wave function of an infinite sheet of charge and pinhole system be solved analytically?

Yes, the quantum wave function for this system can be solved analytically using the Schrödinger equation. However, the calculations can be complex and require advanced mathematical techniques.

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