# Stationary states of free particle

• Mindscrape
In summary, the conversation discusses obtaining the stationary states for a free particle in three dimensions by separating the variables in Schrödinger's equation. The steps involved include substituting the wavefunction into the time-dependent Schrödinger equation, dividing by the wavefunction, and using the separation of variables method to obtain three separate 1-dimensional Schrodinger equations. The conversation also mentions the need for energy in solving the equations.
Mindscrape
The problem is to obtain the stationary states for a free particle in three dimensions by separating the variables in Schrödinger's equation.

So take

$$\psi(\mathbf{r},t) = \psi_1(x) \psi_2(y) \psi_3(z) \phi(t)$$

and substitute it into the time-dependent Schrödinger equation. For the stationary states set U=0 and obtain

$$\frac{-\hbar^2}{2m} \nabla^2 \psi(\mathbf{r},t) = i\hbar \frac{\partial \psi(\mathbf{r},t)}{\partial t}$$

Then divide by the wavefunction, and I get

$$i\hbar \frac{\partial \phi(t)}{\partial t} = \frac{-\hbar^2}{2m} \left( \frac{\partial^2 \psi_1}{\partial x^2} + \frac{\partial^2 \psi_2}{\partial y^2} + \frac{\partial^2 \psi_3}{\partial z^2} )\right$$

I know that each one of the unknown functions must make a separate equation, but I don't know what to solve for without energy. For the time-independent equation they will all essentially be infinite square wells, but I don't know what to do with the time dependency.

Mindscrape said:
The problem is to obtain the stationary states for a free particle in three dimensions by separating the variables in Schrödinger's equation.

So take

$$\psi(\mathbf{r},t) = \psi_1(x) \psi_2(y) \psi_3(z) \phi(t)$$

and substitute it into the time-dependent Schrödinger equation. For the stationary states set U=0 and obtain

$$\frac{-\hbar^2}{2m} \nabla^2 \psi(\mathbf{r},t) = i\hbar \frac{\partial \psi(\mathbf{r},t)}{\partial t}$$

Then divide by the wavefunction, and I get

$$i\hbar \frac{\partial \phi(t)}{\partial t} = \frac{-\hbar^2}{2m} \left( \frac{\partial^2 \psi_1}{\partial x^2} + \frac{\partial^2 \psi_2}{\partial y^2} + \frac{\partial^2 \psi_3}{\partial z^2} )\right$$

I know that each one of the unknown functions must make a separate equation, but I don't know what to solve for without energy. For the time-independent equation they will all essentially be infinite square wells, but I don't know what to do with the time dependency.

Oops. A correction. I had no noticed that you had put in the wavefunction and then divided by it.

Ok, so at first you should get

$$i\hbar \psi_1 \psi_2 \psi_3 \frac{\partial \phi(t)}{\partial t} = \frac{-\hbar^2}{2m} \left( \phi \psi_1 \psi_3 \frac{\partial^2 \psi_1}{\partial x^2} +\phi \psi_1 \psi_3 \frac{\partial^2 \psi_2}{\partial y^2} + \phi \psi_1 \psi_2 \frac{\partial^2 \psi_3}{\partial z^2} )\right$$

Then the next step is to divide everything by $\phi \psi_1 \psi_2 \psi_3$ and then you should get (instead of what you wrote):

$$i\hbar {1 \over \phi} \frac{\partial \phi(t)}{\partial t} = \frac{-\hbar^2}{2m} \left( {1 \over \psi_1} \frac{\partial^2 \psi_1}{\partial x^2} +{ 1 \over \psi_2} \frac{\partial^2 \psi_2}{\partial y^2} + {1 \over \psi_3} \frac{\partial^2 \psi_3}{\partial z^2} )\right$$
and then to use the usual argument of separation of variables to show that $\phi(t)$ obeys
[tex] i \hbar {\partial \phi(t) \over \partial t} = E \phi(t) [/itex] where E is the constant of separation. So $\phi(t) = A e^{-iE t / \hbar}$.

Then you go on to separate the equations in x, y and z. You end up with three separate 1-dimensional Schrodinger equations ).

Hope this helps.

Patrick

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## What are stationary states of free particle?

Stationary states of free particle refer to the quantum mechanical solutions to the Schrödinger equation for a particle that has no external forces acting upon it. These solutions describe the probability of finding the particle at a particular position and time.

## What is the significance of stationary states of free particle?

Stationary states of free particle are significant because they provide a complete set of solutions to the Schrödinger equation, allowing us to accurately describe the behavior of a free particle. They also form the basis for understanding more complex quantum systems.

## How do stationary states of free particle differ from bound states?

Stationary states of free particle are characterized by the particle having a definite energy, but can have a range of positions and momenta. Bound states, on the other hand, have the particle confined to a specific region of space due to external forces, and therefore have a discrete set of energies.

## What are the possible values for the energy of a stationary state of free particle?

The energy of a stationary state of free particle can take on any value, as it is not restricted by external forces. However, the probability of finding the particle at a higher energy is lower than finding it at a lower energy, due to the nature of the solutions to the Schrödinger equation.

## How do stationary states of free particle relate to the uncertainty principle?

The uncertainty principle states that it is impossible to know both the position and momentum of a particle with absolute certainty. Stationary states of free particle are a manifestation of this principle, as they describe the probability of finding a particle at a certain position and time, but do not specify its exact position and momentum.

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