Finding the eigen function for an infinite square well (quantum mechanics)

In summary, the conversation discusses a problem in quantum mechanics where a proton is confined in an infinite square well and undergoes a transition from the first excited state to the ground state. The question asks for the calculation of the associated energy and wavelength of the emitted photon, as well as the region of the electromagnetic spectrum it belongs to. The solution involves using the time independent Schrodinger equation and solving for the wavefunction using the given potential. The wavefunction must vanish at the boundaries of the well, which leads to restrictions on the possible values of k. The conversation ends with the speaker questioning their understanding of the problem.
  • #1
ElijahRockers
Gold Member
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Homework Statement



Quantum mechanics is absolutely confusing me.

A proton is confined in an infinite square well of length 10-5nm.

Calculate the wavelength and energy associated with the photon that is emitted when the proton undergoes a transition from the first excited state (n=2) to the ground state (n=1).

In what region of the electromagnetic spectrum does this wavelength belong?

The Attempt at a Solution



I'm not really sure what I'm doing at all.

But I started with the time independent Schrodinger equation. For region I and III, (where the potential is infinite) then the eigenfunction must be 0.

So I put in a potential of zero for the second region and got

[itex]\frac{d^2\psi}{dx^2} + k^2\psi = 0[/itex] where [itex]k^2 = \frac{2mE}{\hbar}[/itex]

Looks like a second order diff eq, so I tried to solve it. Solutions to the characteristic equation were ±ik...

From here I am a little stumped, I didn't take notes, and I can't remember what the solution was that he used in class.

But anyway if I'm just going by my normal diff eq understanding,

[itex]\psi = Acos(kx)+Bsin(kx)[/itex]

What's throwing me off is that I recall he had exponential solutions with imaginary components in them. These would oscillate, and so does my solution, but even if I had my solution in his form, what next?

Where does the length of the well come into play? What does n=2 -> n=1 mean?
Am I making this more complicated than it should be?
 
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  • #2

1. What is an eigen function in quantum mechanics?

An eigen function is a wave function that describes the energy states of a quantum system. It is a solution to the Schrödinger equation and represents the probability amplitude of finding a particle in a specific energy state.

2. What is an infinite square well potential?

An infinite square well potential is a hypothetical potential energy profile in which a particle is confined to a specific region, with an infinitely high potential energy barrier at the boundaries. This type of potential is commonly used to model quantum systems, such as a particle in a box.

3. How do you find the eigen function for an infinite square well potential?

To find the eigen function for an infinite square well potential, you need to solve the Schrödinger equation for the given potential. This involves applying boundary conditions and using mathematical techniques, such as separation of variables, to find the possible energy states and corresponding eigen functions.

4. Why is the eigen function for an infinite square well potential important?

The eigen function for an infinite square well potential is important because it allows us to understand the energy states and behaviors of a quantum system. It can also be used to calculate the probability of finding a particle in a specific energy state, which is crucial for understanding the behavior of quantum particles.

5. Can the eigen function for an infinite square well potential be used for other quantum systems?

Yes, the eigen function for an infinite square well potential can be used for other quantum systems with similar boundary conditions. However, the specific shape and parameters of the potential may vary, and the eigen functions and energy states will be different for each system.

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