# Probability density from Wave Function

• Mandelbroth
In summary, the conversation discusses the disagreement between two individuals regarding the probability density of a particle in space. While one believes it is given by the square of the wave function, the other argues that it is given by the wave function multiplied by its complex conjugate. The summary concludes that the general expression for the probability density is given by the absolute value of the wave function squared, while noting that there may be exceptions in certain cases.
Mandelbroth
A friend of mine recently tried to tell me that the square of the wave function for a particle (that is, $\Psi^2$) gives the probability density of finding a particle in space.

I disagree. I always thought that the wave function multiplied by its complex conjugate (that is, $\Psi \Psi^*$) yielded the probability density for the particle. They are definitely not the same, because $\forall a,b \neq 0, \ (a+bi)^2 = a^2 + 2abi + b^2 \neq a^2 + b^2$.

So, is the probability density given by $\Psi^2$ or $\Psi \Psi^* = |\Psi|^2$?

Mandelbroth said:
A friend of mine recently tried to tell me that the square of the wave function for a particle (that is, $\Psi^2$) gives the probability density of finding a particle in space.

I disagree. I always thought that the wave function multiplied by its complex conjugate (that is, $\Psi \Psi^*$) yielded the probability density for the particle. They are definitely not the same, because $\forall a,b \neq 0, \ (a+bi)^2 = a^2 + 2abi + b^2 \neq a^2 + b^2$.

So, is the probability density given by $\Psi^2$ or $\Psi \Psi^* = |\Psi|^2$?

It's the $|\Psi|^2$, a real number

Perhaps your friend referred to wave function that is eigenstate of some atomic or molecular Hamiltonian. These can be chosen to be real, so then ##\Psi^2## gives the correct density as well as ##|\Psi|^2##. But the latter is the general expression.

## What is a probability density function (PDF)?

A probability density function, or PDF, is a mathematical function that describes the likelihood of a random variable having a certain value within a given range. It is often used to analyze the probability of events in statistics and probability theory.

## How is a probability density function related to a wave function?

A wave function is a mathematical description of a quantum system, typically used to determine the probability of finding a particle at a specific location. The probability density function can be derived from the square of the wave function, giving the probability of finding a particle at a specific location in space.

## How is the probability density function used in quantum mechanics?

In quantum mechanics, the probability density function is used to determine the likelihood of finding a particle at a specific position in space. It is also used to calculate the expectation value of a given observable, such as energy or momentum.

## What is the difference between a probability density function and a probability distribution?

A probability density function describes the probabilities of a continuous random variable, while a probability distribution describes the probabilities of a discrete random variable. In other words, a probability density function is used for continuous data, while a probability distribution is used for discrete data.

## Can a probability density function have negative values?

No, a probability density function cannot have negative values. The total area under the curve of a probability density function must equal 1, and since probability cannot be negative, the function must also be non-negative.

Replies
9
Views
853
Replies
9
Views
1K
Replies
21
Views
1K
Replies
5
Views
911
Replies
1
Views
819
Replies
25
Views
2K
Replies
17
Views
2K
Replies
23
Views
2K
Replies
8
Views
1K
Replies
6
Views
2K