Stationary states of free particle

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.
  • #1
Mindscrape
1,861
1
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

[tex]\psi(\mathbf{r},t) = \psi_1(x) \psi_2(y) \psi_3(z) \phi(t)[/tex]

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

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

Then divide by the wavefunction, and I get

[tex]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 [/tex]

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.
 
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  • #2
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

[tex]\psi(\mathbf{r},t) = \psi_1(x) \psi_2(y) \psi_3(z) \phi(t)[/tex]

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

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

Then divide by the wavefunction, and I get

[tex]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 [/tex]

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



[tex]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 [/tex]

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

[tex]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 [/tex]
and then to use the usual argument of separation of variables to show that [itex] \phi(t) [/itex] obeys
[tex] i \hbar {\partial \phi(t) \over \partial t} = E \phi(t) [/itex] where E is the constant of separation. So [itex] \phi(t) = A e^{-iE t / \hbar} [/itex].

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
 
Last edited:

Related to Stationary states of free particle

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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