# Potential barrier, wave function, and probability

• judonight
In summary: It's the only thing that's observable, so it's all you need to know. The time-dependence of the real and imaginary parts of the probability density are dictated by the time-dependent Schrodinger equation as well.
judonight
I have a couple questions about finite potential barriers that I can't seem to figure out on my own...

1) Why does the real part of the wave function collapse inside the barrier (square, rectangular, barrier with V less than the energy of particle)? It seems to me that there should be some probability that the particle, if tunneling, could get trapped inside the barrier, since as the wave collides with the barrier the wave function is non zero.

2) Is the wave function 'expanding' after 'collision' witht the barrier (for the reflected and transmitted waves) because the probability of it's position is changing? Has it lost momentum?

3) How is the real and imaginary components of the probability density dependant on time after the 'collision'? Is it more than just following the wave function? I understand that the amplitude is ever lowering due to the 'expansion' of the wave packets corresponding to the transmitted and reflected waves, but, not sure how each component are dependant on time.

I know these aren't great questions, but in my attempt to understand the quantum world these few things are still bothering me.

1) The particle will not be trapped in the barrier if its energy is greater than the barrier potential ##V##. You are right, the wave function is non-zero there so there is some probability that it will be found there, just as there is everywhere else that the decaying exponential tail of the wave packet reaches - but of course it is much more likely to be found in the high-amplitude areas of the reflected and transmitted wave packets.

2) Clearly if the particle is detected in the reflected wave its momentum will have changed; the discrepancy is made up by a change in the momentum whatever is creating the barrier (which is assumed to be so massive that the effect of the momentum transfer on it can be ignored). This is the same thing we do with momentum conservation in classical physics when bouncing a ball off an immovable wall.
The wave packet is always "expanding". It's a superposition of many different frequencies so it spreads out over time. This is all determined by the time-dependent Schrodinger equation. (It's also computationally a nuisance, so when we just need the probabilities of transmission and reflection we work with a plane wave of a given energy instead of a wave packet).

3) Yes, you just follow the wave function.

## What is a potential barrier?

A potential barrier is an energy barrier that exists between two regions in a system. It prevents particles or waves from moving freely between the two regions and can be created by a variety of factors, such as electric fields, magnetic fields, or changes in the physical properties of the system.

## What is a wave function?

A wave function is a mathematical representation of a quantum mechanical system. It describes the probability of finding a particle or wave at a particular location in space and time. The square of the wave function gives the probability density, which represents the likelihood of finding the particle at a specific point.

## How is probability related to wave functions?

The probability of finding a particle or wave at a specific location is directly related to the square of the wave function. This means that the higher the probability density at a certain point, the more likely it is to find the particle or wave at that location. Probability is a fundamental concept in quantum mechanics and is described by the wave function.

## What is the Schrödinger equation and how does it relate to wave functions?

The Schrödinger equation is a mathematical equation that describes how the wave function of a quantum system changes over time. It is a fundamental equation in quantum mechanics and is used to calculate the possible states of a system and the probabilities of those states. The wave function is a solution to the Schrödinger equation, and its evolution over time can be described by this equation.

## How does a potential barrier affect the wave function and probability of a particle?

A potential barrier can cause the wave function of a particle to change as it passes through it. This change in the wave function can result in a change in the probability of finding the particle in different locations. In some cases, the potential barrier can completely reflect the particle, meaning that the probability of finding it on one side of the barrier is zero. In other cases, the particle may have a non-zero probability of tunneling through the barrier, resulting in a non-zero probability of finding it on the other side.

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