Quantum Mechanics: Choose an acceptable bound state function

• 511mev
In summary, there are certain criteria for a bound state wave function to be acceptable, such as going to zero at infinity, being smooth and continuous, and belonging to a properly chosen L^2 space. The given options (1), (3), and (4) all have their own unique properties, but (2) does not satisfy the criteria and is not a valid solution.
511mev
1. Which of the following is an allowed wave function for a particle in a bound state? N is
a constant and α, β>0.
1) Ψ=N e-α r
2) Ψ=N(1-e-α r)
3) Ψ=Ne-α x e-β(x2+y2+z2)
4) Ψ=Non-zero constant if r<R , Ψ=0 if r>R

Only one is correct.

2. What are the criteria for acceptable bound state wave functions?

3. I did assume that one of the criteria is that the wave function must go to zero at infinity. To show this, I took the limits as r goes to infinity and, for the function given in 3, as x,y,z, go to positive and negative infinty. I got that all are zero at infinity except 2.
Another requirement is that the function be smooth and continuous. Since 4 has an abrupt change, i.e., its derivative is infinite at R, then it is not smooth.
That leaves 1 and 3. What additional property am I forgetting?

It seems to me the problem is wrong in claiming only one answer is correct. The wave functions in (1) and (3) are solutions to, respectively, the hydrogen atom and the simple harmonic oscillator, for the appropriate choices of α and β.

511mev said:
[...] I did assume that one of the criteria is that the wave function must go to zero at infinity. [...]

This is wrong. The requirement is being a member of a properly chosen L^2.

Are you saying that my assertion that a bound state wave function go to zero at infinity is incorrect? Would not a member of a well chosen L^2 have this property if it were to represent a bound state?

That is what I thought. I am attempting to help a student with this problem on Cramster. He claims that there is only one answer. I cannot formulate a proof, though.

I'm saying that there are wavefunctions which don't go to 0 when approaching infinity, but are still square integrable. But such <oddities> can be the limits of convergent sequences of <good> wavefunctions (=those who have the particular infinity behavior). In the language of functional analysis, S(R) is a proper subset of L^2(R) and dense everywhere in it.

Anyways, regarding the problem itself, as unveiled above, both (1) & (3) are valid solutions to your problem.

1. What is a bound state function in quantum mechanics?

A bound state function in quantum mechanics is a mathematical representation of a particle or system that is confined to a specific energy state. This function describes the probability of finding the particle in a particular location within the system.

2. How is a bound state function chosen in quantum mechanics?

The bound state function is chosen based on the properties of the system being studied. It must obey certain mathematical conditions, such as being continuous and finite, and must be normalized so that the total probability of finding the particle is equal to 1.

3. What makes a bound state function acceptable in quantum mechanics?

An acceptable bound state function must be able to accurately describe the behavior of the particle or system in question. This means that it must satisfy the Schrodinger equation and accurately predict the energy levels and probabilities of the particle or system.

4. Can a bound state function change over time in quantum mechanics?

In most cases, a bound state function will not change over time in quantum mechanics. However, in certain situations, such as when the system is subjected to external forces or interactions, the bound state function may evolve or change to reflect these changes in the system.

5. How does the choice of bound state function affect the results of quantum mechanics experiments?

The choice of bound state function can greatly affect the results of quantum mechanics experiments. A poorly chosen function may not accurately describe the behavior of the system, leading to incorrect predictions and results. Therefore, it is important to carefully select an appropriate bound state function for each system being studied.

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