How Does the Wave Function Behave Across a Step Potential in Quantum Mechanics?

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Schrodinger equation for a particle moving through a step potential and has asked for clarification on a previous thread.In summary, the conversation revolves around a particle with mass m and energy E moving in one dimension from right to left, incident on a step potential with V(x)=0 for x<0 and V(x)=Vo for x>=0, where Vo>0. The energy of the particle is E>Vo. The Schrodinger equation needs to be solved to derive the wave function for x<0 and x>=0, and the solution can be expressed in terms of a single unknown constant. The conversation also touches on the matching of waves in different zones and the boundary conditions for their am
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abubabu
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A particle with mass m and energy E is moving in one dimension from right to left. It is incident on the step potential V(x)=0 for x<0 and V(x)=Vo <--(pronounced "v not") for x>=0, where Vo>0, as shown on the diagram. The energy of this particle is E>Vo.
(The diagram has a particle coming in from the left above V=Vo)

Solve the Schrodinger equation to derive the wave function for x<0 and x>=0. Express the solution in terms of a single unknown constant.

I have been looking at the case when a particle is coming in from the right and E>Vo, but I am failing to make a connection between the two. I really need the idea explained to me here, I'm missing something!
 
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I just wrote up an answer to almost the same question in another thread with almost the same name as this one: "1-dimensional time-independent Schroedinger equation"

The question in the other thread was messed up because it put in a second step potential; but essentially you didn't need to worry about that in the end. Here is my write-up from the other thread:

I can't read the correction you made to your original post but the problem only makes sense to me if there is an incoming particle (wave actually) from the left, a step in potential at x=0 and another one at x=a. Then there is a reflection at the first boundary, a middle zone with waves going both ways, and a transmitted beam at the second boundary.

I think you already have expressions for the waves, which is to say their k numbers, based on the three potentials. So there are 5 undetermined coefficients, in general complex, for the wave amplitudes.

However, as I see it, it gets a lot simpler when you consider what they are asking for: just the ratio between the left- and right- propagating waves in the middle zone. Let's take the outgoing wave in zone 3 to have unit amplitude; in fact, make the wave number something simple like 1 so it is just:

exp(ikx - wt)

(of course we won't worry to much about the wt).

Then you just need to match up the waves in zone 2, let's give them a wave number like 3 or something:

Aexp(i3x) + Bexp(-i3x)

And I believe the boundary condition is that both the amplitudes and their derivatives have to match up at the transition. With the arbitrary numbers I put in above, I can solve pretty easily: I get A + B = 1 for the amplitudes, and 3A - 3B = 1 for the derivatives.

So it seems I can solve the problem as stated without even worrying about what happened at the first boundary, with the incoming particle. Does this look right?

Marty
 

Related to How Does the Wave Function Behave Across a Step Potential in Quantum Mechanics?

1. What is the one-dimensional time-independent Schrodinger equation?

The one-dimensional time-independent Schrodinger equation is a mathematical equation used in quantum mechanics to describe the behavior of a quantum system in one dimension. It is a second-order partial differential equation that relates the time-independent wave function of a system to its energy.

2. What does the one-dimensional time-independent Schrodinger equation tell us about a quantum system?

The one-dimensional time-independent Schrodinger equation gives us information about the energy levels and wave function of a quantum system. It allows us to calculate the probability of finding a particle in a certain location and understand the behavior of the system over time.

3. How is the one-dimensional time-independent Schrodinger equation derived?

The one-dimensional time-independent Schrodinger equation is derived using the principles of quantum mechanics and the Hamiltonian operator. It is based on the assumption that the wave function of a system can be described by a time-independent equation and that the total energy of the system is equal to the sum of its kinetic and potential energies.

4. What is the significance of the one-dimensional time-independent Schrodinger equation in quantum mechanics?

The one-dimensional time-independent Schrodinger equation is a fundamental equation in quantum mechanics and is used to solve many important problems in the field. It allows us to understand the behavior of quantum systems and make predictions about their properties.

5. How is the one-dimensional time-independent Schrodinger equation solved?

The one-dimensional time-independent Schrodinger equation can be solved using various mathematical techniques, such as separation of variables, perturbation theory, and variational methods. The solutions provide information about the energy levels and wave function of a quantum system.

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