# Difference between expectation value and probabilty

• MGWorden
In summary, the conversation discusses the difference between finding the probability of momentum and the expected value of momentum. The expected value of momentum is the most likely momentum value in a given state, while the probability of momentum is related to the normalized wavefunction in momentum space. To find the probability of a momentum range, integration is necessary.
MGWorden

## Homework Statement

Psi(x) = Ax -a<x<a

I am trying to find the probability that my measured momentum is between h/a and 2h/a

## Homework Equations

I have normalized A= sqrt(3/(2a^3))

I know that if I was finding the expected momentum I would use
$$\int$$$$\Psi$$ * p $$\Psi$$ dx

## The Attempt at a Solution

so far I have done
$$\int$$ $$\Psi$$ * p $$\Psi$$ dx
with bounds from from h/a and 2h/a

but I know this is the expected value of momentum and I don't think that is the same thing as the probability of momentum.

Could someone please explain the difference in finding the probability of momentum and the expected value of momentum. Or if they are the same thing let me know there is no difference.

The probability that the momentum lies between p1 and p2 is not related to the expected value of momentum. The expected value of momentum is the most likely momentum value we'd find in the state $$\Psi(x)$$. The probability to find the particle with a given momentum p is $$| \psi(p) |^2$$, where $$\psi(p)$$ is the normalized wavefunction in momentum space. To find the probability of finding the momentum in a range of values, we have to integrate this.

Thanks for your help. I converted over to momentum space using a Fourier transform then integrated.

Your explanation saved me much time searching.

Cheers!

## What is the difference between expectation value and probability?

Expectation value and probability are two important concepts in statistics and probability theory. While they both relate to the likelihood of an event occurring, they have distinct meanings and applications.

## What is expectation value?

Expectation value, also known as the mean or average, is a mathematical concept that represents the average outcome of a random variable over a large number of trials. It is calculated by multiplying each possible outcome by its respective probability and summing them together.

## What is probability?

Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 represents impossibility and 1 represents certainty. In the context of statistics, probability is used to describe the chance of a particular outcome or event happening.

## How are expectation value and probability related?

Expectation value and probability are related in that they both involve the likelihood of an event occurring. However, they differ in their interpretations and calculations. While probability is a single number that represents the chance of an event happening, expectation value takes into account all possible outcomes and their corresponding probabilities.

## When should expectation value be used instead of probability?

Expectation value should be used when there are multiple possible outcomes with different probabilities. It is particularly useful in decision-making and risk assessment, as it provides a more comprehensive understanding of the potential outcomes of a situation. On the other hand, probability is more suitable for simple events with only two possible outcomes.

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