# Probability dependence on potential

• Kaguro
In summary, the conversation discusses the relationship between potential energy and frequency in the Schrodinger equation, as well as the behavior of amplitude and frequency in different potential energy scenarios for the harmonic oscillator and finite square well. The conversation also mentions a homework question about choosing the correct probability density given a potential energy graph, and provides a hint to help with the selection.
Kaguro
Homework Statement
Say, I am given a random looking graph of potential V(x) vs x. I'm not given any information about particle energy. The question then asks me to choose the correct option of ##|\psi (x)|^2##. I feel like the probability density should be higher where potential is lower. But why exactly should that be?
Relevant Equations
I don't know...
I only know that if E>V, then the frequency would be higher where E-V is higher. But what does that have anything to do with probability?

Shrodinger equation is written as
$$[(\frac{d}{dx})^2+\frac{(E-V)2m}{\hbar^2}]\psi=0$$
Depending on sign of E-V ##\psi## does sinusoidal vibration or exponential dumping.

Yes, I know. I wanted to ask if V is relatively lower at an x, would that imply that the amplitude will be relatively higher?

In order to check your hypothesis why do not you review the cases you well know, e.g. V(x) = kx^2, V(x)=0 for -a<x<a, V otherwise, not only ground state but also excited states ?

For Harmonic oscillator, the ground state is fine. But in the excited states, we can see a piling away from centre.

In the finite square well, if E>V, then the amplitude doesn't change, but frequency decreases in the regions |x|>a.

If E<V, then it shows an exponentially decaying amplitude.

If there's a block of V, then after exponentially decaying, it again becomes simple harmonic with reduced amplitude.

Your consideration seems reasonable to me, though I am not confident in catching your teacher's intention.

Kaguro said:
Homework Statement:: Say, I am given a random looking graph of potential V(x) vs x. I'm not given any information about particle energy. The question then asks me to choose the correct option of ##|\psi (x)|^2##.
What do you mean by "choose the correct option"? Are you given a certain number of options and are asked to pick one or more? If so, what are they?

The potential is like this:

The probability densities are like this:

Something like this. c and d are useless. a and b.

a is similar to the potential given. b is the opposite.

Probability densities are square integrable. Is that the case with all the choices? Hint: No.

Of the ones that are square integrable, which one looks like the best candidate given that your potential looks like it does?

Also, don't expect the probability density to be similar to the potential, whatever that means.

## 1. How does potential affect probability?

Potential refers to the energy of a system, and it can affect the probability of a certain event occurring by influencing the behavior of particles within the system. In quantum mechanics, the potential energy of a particle can determine the probability of its position and momentum at any given time.

## 2. Is there a direct relationship between potential and probability?

Yes, there is a direct relationship between potential and probability. In general, a higher potential energy will result in a lower probability of a certain event occurring, while a lower potential energy will result in a higher probability. This can be seen in the Schrödinger equation, where the potential term affects the overall probability distribution of a particle.

## 3. Can potential change the outcome of a probability experiment?

Yes, potential can change the outcome of a probability experiment. In quantum mechanics, the potential energy of a system can affect the probability of a particle being in a certain state or having a certain energy. This means that the potential can alter the expected outcome of a probability experiment.

## 4. How does the shape of the potential energy curve impact probability?

The shape of the potential energy curve can greatly impact probability. In a simple harmonic oscillator, for example, the potential energy curve is parabolic and results in equally spaced energy levels. This leads to a higher probability of finding a particle in certain energy levels compared to others. In more complex systems, the shape of the potential energy curve can greatly influence the probability distribution of particles.

## 5. Can potential affect the probability of a particle tunneling through a barrier?

Yes, potential can affect the probability of a particle tunneling through a barrier. In quantum mechanics, a particle has a certain probability of tunneling through a potential barrier, which is influenced by the height and width of the barrier. A higher potential barrier will result in a lower probability of tunneling, while a lower potential barrier will result in a higher probability.

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