Solve Finite Potential Well: Schrödinger Eqn. & k=qtan(q*a)

In summary, the given equations for the wavefunction and the derived equations for continuity can be reduced to the formula ##k=q*tan(q*a)## when A=0, by dividing the equations and using the fact that B=0.
  • #1
feynwomann
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I've got these solutions to the Schrödinger equation (##-\frac{\hbar} {2m} \frac {d^2} {dx^2} \psi(x) + V(x)*\psi(x)=E*\psi(x)##):
x < -a: ##\psi(x)=C_1*e^(k*x)##
-a < x < a: ##\psi(x)=A*cos(q*x)+B*sin(q*x)##
x > a: ##\psi(x)=C_2*e^(-k*x)##

##q^2=\frac {2m(E+V_0)} {\hbar^2}## and ##k^2=\frac {2mE} {\hbar^2}##

In a previous exercise i showed that in order for the wavefunction to be continuous the following has to be true:
For x = -a: ##C_1*e^-(k*a)=A*cos(q*a)-B*sin(q*a)##
For x = a: ##C_2*e^-(k*a)=A*cos(q*a)+B*sin(q*a)##
and for the derived wavefunctions:
x = -a: ##k*C_1*e^-(k*a)=q*(A*sin(q*a)+B*cos(q*a))##
x = a: ##-k*C_2*e^-(k*a)=-q*(A*sin(q*a)-B*cos(q*a))##

I also know that A*B=0. So here's the question:

I'm asked to show that when A = 0, the four equations above kan be reduced to ##k=q*tan(q*a). But I'm not even sure where to start.
 
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  • #2
feynwomann said:
wjzf9cM.png


'
I've got these solutions to the Schrödinger equation (##-\frac{\hbar} {2m} \frac {d^2} {dx^2} \psi(x) + V(x)*\psi(x)=E*\psi(x)##):
x < -a: ##\psi(x)=C_1*e^(k*x)##
-a < x < a: ##\psi(x)=A*cos(q*x)+B*sin(q*x)##
x > a: ##\psi(x)=C_2*e^(-k*x)##

##q^2=\frac {2m(E+V_0)} {\hbar^2}## and ##k^2=\frac {2mE} {\hbar^2}##

In a previous exercise i showed that in order for the wavefunction to be continuous the following has to be true:
For x = -a: ##C_1*e^-(k*a)=A*cos(q*a)-B*sin(q*a)##
For x = a: ##C_2*e^-(k*a)=A*cos(q*a)+B*sin(q*a)##
and for the derived wavefunctions:
x = -a: ##k*C_1*e^-(k*a)=q*(A*sin(q*a)+B*cos(q*a))##
x = a: ##-k*C_2*e^-(k*a)=-q*(A*sin(q*a)-B*cos(q*a))##

I also know that A*B=0. So here's the question:

I'm asked to show that when A = 0, the four equations above kan be reduced to ##k=q*tan(q*a). But I'm not even sure where to start.

Sure, you do. When [itex]A=0[/itex], your equations become:
  1. ##C_1 e^{-ka}= -B sin(qa)##
  2. ##C_2 e^{-ka}=-B sin(qa)##
  3. ##k C_1 e^{-ka}=q Bcos(qa))##
  4. ##-kC_2*e^{-ka}=q Bcos(qa))##
Using equation 1, you know [itex]C_1e^{-ka} = -Bsin(qa)[/itex]
So replace [itex]C_1 e^{-ka}[/itex] by [itex]-B sin(qa)[/itex] in equation 3.

It doesn't look like it results in the same formula as you are expecting, so maybe there was a mistake somewhere.
 
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  • #3
Oh it totally makes sense now. I forgot that my teacher said there was a mistake in the exercise. It's supposed to be B = 0. I didn't think of dividing the equations (I had actually done the first step, but didn't see how I could take it further). Anyway, thanks for the help!
 
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1. What is a finite potential well?

A finite potential well is a concept in quantum mechanics that describes a finite region in space where a particle is confined due to the presence of a potential barrier. This can be visualized as a particle trapped in a well, where the walls of the well act as a barrier preventing the particle from escaping.

2. What is the Schrödinger equation?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system over time. It is a mathematical equation that relates the energy of a system to its wave function, which represents the probability of finding a particle at a certain location in space.

3. How is the Schrödinger equation used to solve a finite potential well?

The Schrödinger equation is used to solve a finite potential well by using boundary conditions, such as the potential at the boundaries of the well, to find the allowed energy levels and corresponding wave functions for the particle within the well. The solution to the Schrödinger equation gives us information about the behavior and properties of the particle within the well.

4. What is the significance of the parameter k in the equation?

The parameter k in the equation represents the wave number, which is related to the momentum of the particle. It is used to determine the allowed energy levels and wave functions for the particle in the finite potential well. In the context of the equation k=qtan(q*a), it is used to find the allowed values of q, which in turn determine the energy levels and wave functions.

5. How does the value of k affect the behavior of a particle in a finite potential well?

The value of k affects the behavior of a particle in a finite potential well by determining the allowed energy levels and corresponding wave functions for the particle. A higher value of k corresponds to a higher momentum and therefore a greater likelihood of the particle being found at a greater distance from the center of the well. This can also affect the probability of the particle tunneling through the potential barrier of the well.

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