# How Do You Find the Probability Current of a Free Particle?

• petera88
In summary, the conversation discusses finding the probability current of a free particle using the time dependent wave function. The complex conjugate of the wave function is explained and it is noted that when the wave function is real, there is no probability current. The equation for the probability current is provided and it is mentioned that the final answer is not zero.
petera88

## Homework Statement

Find the probability current of a free particle.

## Homework Equations

$\Psi$(x,t) = Aei(kx-$\frac{(hbar)k^{2}t}{2m}$)

J(x,t) = $\frac{ihbar}{2m}(ψψ*' - ψ*ψ')$

## The Attempt at a Solution

I figured it was just take the derivative of the time dependent wave function and plug it in. This is my first experience with quantum mechanics so I find myself getting caught up on working with the wave function. My question it if it's real, then the psi and psi* are the same and it would equal 0. This isn't the answer of course. How do I work with the complex psi? What's the difference in the wave function for psi and psi*?

To answer the simplest question, $\psi^*$ is just the complex conjugate of ψ:

For some complex number $c=a+bi$, $c^*=a-bi$.
For a real number $a$, $a=a^*$
In the case of complex exponential functions:

$c=Ae^{ix}$, $c^*=A^*e^{-ix}$

Finally, $cc^*=c^*c=|c|^2$, which is a positive real number.

The equation for a free particle in one dimension is

$\psi(x,t) = Ae^{ikx}e^{-i\frac{\hbar k^2}{2m} t}$

Its complex conjugate is

$\psi^*(x,t) = A^*e^{-ikx}e^{i\frac{\hbar k^2}{2m} t}$

Note that by writing this as the wave function of the particle your starting with the assumption that $\psi$ is complex; it almost always IS complex, with a few exceptions (such as the energy eigenstates the particle in a box. When a particle's wave function is real there is 0 probability current; the probability distribution of the particle does not evolve in time (though its wave function still does). You work with complex $\psi$ like any other function, just making sure to due the complex arithmetic correctly; the derivitives all behave the same way as a real valued function.

So, back to the problem:

The probability current is defined as:
$J(x,t)=\frac{i\hbar}{2m}(\psi\frac{∂\psi^*}{∂x}-\psi^*\frac{∂\psi}{∂x})$

Just take derivitives like you would normally; I will say that the final answer is not zero.

## 1. What is a probability current free particle?

A probability current free particle is a theoretical concept in quantum mechanics that describes a particle moving through space with a constant probability of being found at any given point. This means that the particle does not have a specific trajectory or location, but is instead described by a probability distribution.

## 2. How is the probability current of a free particle calculated?

The probability current of a free particle is calculated by taking the product of the particle's probability density and its velocity at a given point in space. This gives the rate at which the probability of finding the particle changes over time at that specific point.

## 3. Can a particle have a nonzero probability current?

Yes, a particle can have a nonzero probability current if it is under the influence of an external force or potential. In this case, the particle's probability distribution and velocity will change over time, resulting in a nonzero probability current.

## 4. What is the significance of a probability current free particle in quantum mechanics?

The concept of a probability current free particle is important in understanding the fundamental principles of quantum mechanics, such as the uncertainty principle and wave-particle duality. It also helps to explain the behavior of particles in quantum systems and their interactions with external forces.

## 5. Are probability current free particles observed in real-world systems?

No, probability current free particles are a theoretical concept and have not been observed in real-world systems. However, they serve as a useful tool for understanding the behavior of particles in quantum mechanics and can be used to make accurate predictions about the behavior of quantum systems.

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