Particle moving in potential

In summary: An integral from 0 to ∞ of the modulos squared of the wavefunction must be equal to 1. That's what normalization means.In summary, the wave function for a particle of mass m moving in the potential V = { ∞ for x=0 { 0 for 0 < x ≤ a { V0 for x ≥ a is ψ(x) { Asin(kx) for 0 < x ≤ a { Ce-Kx for x ≥ a, with k = √[(2mE/h2)] and K = √[((2m(V0-E)/h2)], where h is h-bar in both equations, and remains so throughout this thread. Applying the boundary conditions at x = a
  • #1

Homework Statement


The wave function for a particle of mass m moving in the potential
V =
{ ∞ for x=0
{ 0 for 0 < x ≤ a
{ V0 for x ≥ a

is
ψ(x)
{ Asin(kx) for 0 < x ≤ a
{ Ce-Kx for x ≥ a

with

k = √[(2mE/h2)]

K =√[((2m(V0-E)/h2)]

where h is h-bar in both equations, and remains so throughout this thread.

(a) Apply the boundary conditions at x = a and obtain the transcendental equation which determines the bound state energies E.
(b) If
√[(2mV0a2)/h2] = 3pi

determine the allowed bound state energies. Express your answers in the form of a numerical factor multiplying the dimensional factor (h2/(2ma2)).

(c) Given that the normalization factor for the wave function corresponding to the nth energy En is

An = √[2/a]*√[(Kna)/(1+(Kn*a))]

normalize the wave function for the lowest energy state.
(d) What is the probability that a measurement of the position of a particle in the ground state will give a result ≥ a?

Homework Equations





The Attempt at a Solution


I have completed parts (a) and (b) and I believe they are correct, here is a brief outline of what I did.
At x = a: ψ(a)

Asin(ka) = Ce-Ka (1)

dψ/dx

kAcos(kx) = -KCe-Ka (2)

Dividing (2) by (1):

kcot(ka) = -K
For the transcendental: z = ka, z0 = a√[(2mV0)/h2]

∴ -cot(z) = √[(z0/z)2-1]

Now setting z0 = 3pi (part b) and graphing -cot(z) and √[(z0/z)2-1] on the same plot, I find that intersections occur just below zn = npi

z = ka = a√[(2mE/h2)] ≈ npi

∴ Solving for En = (n2pi2h2)/(2ma2)

This is where I hit a road block. I believe the lowest energy state is E1 because it is just below pi (or 1pi), but I'm not sure what I'm supposed to do with this. I want to plug this into Kn, but that creates quite the mess. As for the second portion of ψ(x), I can just normalize that and solve for C using Kn, and I think I have a handle on part (d) as well. Any hints on part (c) or just checking my work so far would be much appreciated.

 
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  • #2
I believe the lowest energy state is E1 because it is just below pi (or 1pi)
You mean it is the first non-zero energy eigenstate? ##\small E_0=0## and all.

You have the equations for the wavefunctions, you have decided that n=1 is the lowest one, so the lowest energy wavefunction must be ##\psi_1## - which you found an expression for in part a and b.

Part c asks for the normalized wavefunction - which requires the normalization factor A1: which you have an equation for.

What's the problem?

You could always check by normalizing the hard way.
 

1. What is a particle moving in potential?

A particle moving in potential refers to a physical system in which a particle experiences a force due to an external potential. This potential can be caused by factors such as electric or magnetic fields, gravity, or interactions with other particles.

2. How is the motion of a particle in potential described?

The motion of a particle in potential is described using classical mechanics or quantum mechanics, depending on the scale of the system. In both cases, the motion is described by equations of motion that take into account the force from the potential and the initial conditions of the particle.

3. What is the difference between a conservative and non-conservative potential?

A conservative potential is one in which the work done by the force on the particle depends only on the initial and final positions of the particle and not on the path taken. In contrast, a non-conservative potential results in a nonzero work done on the particle along any path between the same two points.

4. How does a particle's energy relate to its motion in potential?

A particle's energy is directly related to its motion in potential. In a conservative potential, the total energy (kinetic + potential) of the particle remains constant. In a non-conservative potential, the total energy of the particle can change due to the work done by the non-conservative force.

5. What are some real-life examples of a particle moving in potential?

There are many examples of particles moving in potential in everyday life. Some common examples include a charged particle moving in an electric field, a ball rolling down a hill due to gravity, or a pendulum swinging back and forth due to the force of gravity. In physics and chemistry, particles moving in potential are also used to describe the behavior of atoms, molecules, and subatomic particles.

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