Time-independent wave function formula

In summary, the student attempted to solve a problem about obtaining energies in a 1-D box from 0 to a by constructing a wavefunction with given energies and probabilities.
  • #1
droedujay
12
0

Homework Statement



Construct wavefunction with given energies and probabilities of obtaining energies in a 1-D box from 0 to a

Homework Equations


3. The Attempt at a Solution

I know the general form of a time-independent wavefunction but I don't know what to do with the probabilities of obtaining energies. Is there a formula for this?
 
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  • #2
There are lots of places you can find detailed explanations of this problem - it is one of the most common quantum models out there. Try doing a Google search for "infinite square well."
 
  • #3
I looked but I could not find anything on probabilities of obtaining Energies. I know about psi star psi being the probability of finding a particle in a specific region but I do not have any material on probability of obtaining Energies.
 
  • #4
droedujay said:
I know about psi star psi being the probability of finding a particle in a specific region

My quantum is a little shaky, but I think [tex]<\psi_{n}^{*}|\psi_{n}>[/tex] is indeed the position operator, which is also written [tex]|\psi_{n}(x)|^{2}[/tex]. You need to normalize your psi functions so that each [tex]\psi_{n}(x)[/tex] has the correct coefficient: since the particle has to be somewhere in the box, you know that the integral over the whole region must be 1. To find the probability of a certain state, you put the Hamiltonian operator into the Bra-Ket: [tex]<\psi_{n}^{*}|\hat{H}|\psi_{n}>[/tex] So once you have the correct coefficients for your set of [tex]\psi_{n}(x)[/tex] functions, you can do [tex]<\psi_{n}^{*}|\hat{H}|\psi_{n}>[/tex] for each one to find the probability of the particle being in that state, with that energy.
 
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  • #5
I found out the Coefficient expansion theorem and constructed the following wavefunction:

Ψ(x,0) = 1/sqrt(2)*φ1 + sqrt(2/5)*φ3 + 1/sqrt(10)*φ5
where φn = sqrt(2/a)*sin(n*pi*x/a)

Is this unique why or why not? I'm thinking that it has something to do with all odd Energies.
 
  • #6
Beyond what I wrote before I am out of my depth, but I'll try. I believe the energies are odd because the well runs from 0 to a and not -a/2 to a/2; in the latter case I think you would have even energies or cosines. As to the uniqueness question, I would guess that the wavefunction itself is a unique solution to your particular square well, but it is of course a linear combination of eigenfunctions so I don't know how that affects uniqueness. I would imagine performing the Hamiltonian on completely different wavefunctions could return the same energies, so the energy values would not necessarily be unique. Is that what you mean?
 
  • #7
I think that it has something to do with the fact that this wavefunction is orthonormal.
 
  • #8
I believe the orthonormal bit refers to the fact that each eigenfunction that makes up your wavefunction (i.e. [tex]\psi_{1} , \psi_{3} , \psi_{5} , [/tex] etc.) is linearly independent of each of the others, and therefore said to be orthonormal, in the same way that the X, Y and Z-axes are independent and orthonormal, for example.
 
  • #9
I appreciate all the help on this problem. I think I got this one down. Can you check out my other forum "QM wavefunctions" and see if you could help out there too.
 

FAQ: Time-independent wave function formula

What is the time-independent wave function formula?

The time-independent wave function formula is a mathematical expression used to describe the behavior of a quantum mechanical system. It represents the probability amplitude of a particle being in a particular state at a given position and time.

What is the significance of the time-independent wave function formula in quantum mechanics?

The time-independent wave function formula is a fundamental concept in quantum mechanics as it allows us to understand and predict the behavior of particles on a microscopic scale. It plays a crucial role in determining the energy levels and properties of quantum systems.

How is the time-independent wave function formula derived?

The time-independent wave function formula is derived from the Schrödinger equation, which is a differential equation that describes the evolution of a quantum system over time. By solving this equation, we can obtain the wave function, which is then used to calculate physical quantities such as energy and probability distributions.

What are the limitations of the time-independent wave function formula?

The time-independent wave function formula is only applicable to non-relativistic systems and does not take into account the effects of special relativity. It also assumes that the potential energy is independent of time, which may not always be the case.

How is the time-independent wave function formula used in practical applications?

The time-independent wave function formula is used in various practical applications, such as in predicting the behavior of electrons in atoms, understanding the properties of molecules, and developing quantum technologies. It is also used in fields like chemistry, materials science, and engineering to model and study quantum systems.

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