Applying particle in a box boundaries to an eigenfunction

In summary, when applying the boundary conditions for a particle in a box, using the eigenfunction Ψ(x) = A'eikx + B'e-ikx rather than Ψ(x) = Asinkx + Bcoskx results in a simplified equation Ψ(a) = A'(eika - e-ika). This can be further simplified by expressing eika and e-ika in terms of trigonometric functions, such as sin(ka) and cos(ka).
  • #1
ReidMerrill
66
2

Homework Statement


Use the eigenfunction Ψ(x) =A'eikx + B'e-ikx rather than Ψ(x)=Asinkx + Bcoskx to apply the boundary conditions for the particle in a box. A. How do the boundary conditions restrict the acceptable choices for A’ and B’ and for k? B. Do these two functions give different probability density if each is normalized?

Homework Equations


?

The Attempt at a Solution


I know that the limits are Ψ(0)=Ψ(a)=0, Ψ(x)=0

When I apply that to the equation
Ψ(0) =A'eik0 + B'e-ik0 = A'+B'= 0
and
Ψ(a) =A'eika + B'e-ika

And I don't know what to do from here. Any help would be appreciated
 
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  • #2
ReidMerrill said:
I know that the limits are Ψ(0)=Ψ(a)=0, Ψ(x)=0
Ψ(0)=Ψ(a)=0 is good. But why Ψ(x)=0?

When I apply that to the equation
Ψ(0) =A'eik0 + B'e-ik0 = A'+B'= 0
and
Ψ(a) =A'eika + B'e-ika

And I don't know what to do from here. Any help would be appreciated
Use what you found from Ψ(0) = 0 to simplify Ψ(a)=0.
 
  • #3
TSny said:
Ψ(0)=Ψ(a)=0 is good. But why Ψ(x)=0?Use what you found from Ψ(0) = 0 to simplify Ψ(a)=0.

I've got further since I posted this.
From A'+B'=0
B'=-A'
Ψ(a)=A'(eika-e-ika
eika=e-ika

But I'm stuck again.
 
  • #4
ReidMerrill said:
I've got further since I posted this.
From A'+B'=0
B'=-A'
Ψ(a)=A'(eika-e-ika)
eika=e-ika

But I'm stuck again.
If you know how to plot the complex numbers eika and e-ika in the complex plane, then you should be able to see the condition for them to be equal.

However, a better way is to recall how to write eika and e-ika in terms of the trig functions sin(ka) and cos(ka), or vice versa.
 

Related to Applying particle in a box boundaries to an eigenfunction

1. What is a particle in a box?

A particle in a box is a theoretical concept in quantum mechanics that describes a particle confined to a one-dimensional space with two boundaries. It is used to model the behavior of particles in a potential well, such as an electron in an atom.

2. How are the boundaries of the box applied to an eigenfunction?

The boundaries of the box are applied to an eigenfunction by setting the wave function to zero at the boundaries. This creates a finite well for the particle to exist in and allows for the calculation of the particle's energy levels and corresponding eigenfunctions.

3. What is an eigenfunction?

An eigenfunction is a mathematical function that, when operated on by a linear operator, results in a scalar multiple of itself. In the context of a particle in a box, the eigenfunctions represent the allowed energy states of the particle.

4. How does the size of the box affect the eigenfunctions?

The size of the box affects the eigenfunctions by determining the allowed energy levels of the particle. A larger box will result in a greater number of allowed energy levels, while a smaller box will have fewer allowed energy levels.

5. What is the significance of applying particle in a box boundaries to an eigenfunction?

Applying particle in a box boundaries to an eigenfunction allows for the calculation of the allowed energy levels and corresponding eigenfunctions of a particle in a confined space. This is important in understanding the behavior of particles in potential wells and has implications in various fields, such as quantum mechanics and solid state physics.

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