Schrödinger for a Step

In summary: Schrödinger equation for a proton approaching a potential barrier at x=0 is given by:Ψ(x) = A sin (kx - ϕ)Where A is a constant and k is the wave number given by:k = √(2m(E-U0))/ħAnd ϕ is a phase shift determined by the boundary conditions. In summary, the solution to the one dimensional time independent Schrödinger equation for a proton approaching a potential barrier at x=0 is given by Ψ(x) = A sin (kx - ϕ), where A is a constant and k is the wave number. This
  • #1
chris_avfc
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0

Homework Statement



Proton traveling along the x-axis approaches a potential barrier at x = 0

Height of potential barrier - U0 = 20eV
Proton Velocity = 44km/s = 44,000m/s

Asked to show that the solution to the one dimensional time independent Schrödinger equation is
[itex]\psi[/itex](x) = A sin (kx - ϕ)

Homework Equations



I know that when the potential is constant and
V(x) = V0 < E

That the solution is the one given, but only because I have read it, I'm not sure how to get get there myself.


The Attempt at a Solution



Reading about it, what I can get is maybe you treat it as one end of an infinite well, so at the step/wall the wave function has to = 0.
Then the need for it to be zero at the walls and non zero away from the wall requires the addition of half a period.
Hence the phi.
But other than that I don't really get it, could anyone help?

Cheers
 
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  • #2




Thank you for your question. You are on the right track in thinking about the potential barrier as an infinite well. In order to solve the Schrödinger equation for this system, we need to apply the boundary conditions of continuity and smoothness at the potential barrier. This means that the wave function and its derivative must be continuous at the barrier.

To solve for the wave function, we can use the general solution of the Schrödinger equation for a one-dimensional system, which is given by:

Ψ(x) = Ae^(i(kx-ωt)) + Be^(-i(kx-ωt))

Where k is the wave number and ω is the angular frequency. In order to satisfy the boundary conditions, we can consider two cases: when the proton is approaching the barrier from the left and when it is approaching from the right.

Case 1: Proton approaching from the left (x<0)
In this case, the potential is zero and the wave function is given by:

Ψ(x) = Ae^(ikx) + Be^(-ikx)

Applying the boundary condition that the wave function must be zero at the barrier (x=0), we get:

Ψ(0) = 0 = A + B

Applying the boundary condition that the derivative of the wave function must be continuous at the barrier, we get:

dΨ/dx (0) = 0 = ikA - ikB

Solving these two equations, we get A = B = 0. This means that the wave function is zero at all points when approaching the barrier from the left.

Case 2: Proton approaching from the right (x>0)
In this case, the potential is U0 and the wave function is given by:

Ψ(x) = Ce^(ikx) + De^(-ikx)

Applying the boundary condition that the wave function must be zero at the barrier (x=0), we get:

Ψ(0) = 0 = C + D

Applying the boundary condition that the derivative of the wave function must be continuous at the barrier, we get:

dΨ/dx (0) = 0 = ikC - ikD

Solving these two equations, we get C = D = 0. This means that the wave function is zero at all points when approaching the barrier from the right.

Combining
 

What is Schrödinger for a Step?

Schrödinger for a Step is a mathematical model used in quantum mechanics to describe the behavior of a particle moving in a potential energy well.

Who created the Schrödinger for a Step model?

The Schrödinger for a Step model was created by Austrian physicist Erwin Schrödinger in 1926.

What is the significance of the Schrödinger for a Step model?

The Schrödinger for a Step model is significant because it was one of the first successful attempts to apply quantum mechanics to real-world situations, and it is still used today to understand the behavior of particles in a potential energy well.

How does the Schrödinger for a Step model work?

The Schrödinger for a Step model uses a mathematical equation known as the Schrödinger equation to describe the quantum state of a particle in a potential energy well. It takes into account both the particle's kinetic energy and the potential energy of the well.

What are some applications of the Schrödinger for a Step model?

The Schrödinger for a Step model has many applications, including predicting the behavior of electrons in atoms, understanding the properties of semiconductors, and studying the behavior of particles in quantum computing.

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