Confusing eigensolutions of a wave function

In summary, the conversation discusses a potential cavity with a given eigenfunction of the wave function and a textbook solution with a phase angle and amplitude of 1. The question asks for clarification on the function and its name. The expert explains that with a positive omega, the potential is actually a delta barrier and the thread is closed.
  • #1
Yourong Zang
5
0
Consider a potential cavity
$$V(r)=\begin{cases}\infty, &x\in(-\infty,0]\\\frac{\hslash^2}{m}\Omega\delta(x-a), &x\in(0,\infty)\end{cases}$$
The eigenfunction of the wave function in this field suffices
$$-\frac{\hslash^2}{2m}\frac{d^2\psi}{dx^2}+\frac{\hslash^2}{m}\Omega\delta(x-a)\psi=E\psi$$
A textbook gives the following solution:
$$\psi(x)=\begin{cases}Asin(kx), &x\in(0,a)\\ sin(kx+\phi), &x\in(a,\infty)\end{cases}$$
where
$$k^2=\frac{2mE}{\hslash^2}$$
I can clearly understand the first part but in the second part, why is the amplitude of the function 1 and why is there a phase angle?

And is this wave
$$\psi(x)=\sin(kx+\phi)$$
called something like the "excitation mode"?
 
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  • #2
Hello Yourong Zang, ##\qquad## :welcome: ##\qquad## !

Please post this kind of questions in the homework forum and use the template -- it's mandatory, see guidelines

Your problem is worked out here With a positive omega you have a delta barrier, not a cavity.
 
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  • #3
BvU said:
Hello Yourong Zang, ##\qquad## :welcome: ##\qquad## !

Please post this kind of questions in the homework forum and use the template -- it's mandatory, see guidelines

Your problem is worked out here With a positive omega you have a delta barrier, not a cavity.
Thank you.
 

1. What are eigensolutions of a wave function?

Eigensolutions of a wave function are the possible states that a quantum system can exist in, represented by the eigenvalues and eigenfunctions of the system's Hamiltonian operator.

2. Why are eigensolutions of a wave function confusing?

Eigensolutions of a wave function can be confusing because they involve complex mathematical concepts and can be difficult to visualize in physical terms.

3. How are eigensolutions of a wave function calculated?

Eigensolutions of a wave function are calculated by solving the Schrödinger equation, which describes the time evolution of a quantum system, for the eigenvalues and eigenfunctions of the system's Hamiltonian operator.

4. What is the significance of eigensolutions of a wave function?

Eigensolutions of a wave function are significant because they represent the possible energy states of a quantum system and can be used to predict the behavior of the system.

5. Can eigensolutions of a wave function change over time?

No, eigensolutions of a wave function do not change over time. They represent the stationary states of a quantum system and remain constant unless acted upon by an external force.

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