[Bioinorganic78]

Hello everyone,

I am currently taking physical chemistry 2. I dont know why, or If I have it right but am just overthinking it, but I am having a hard time visualizing/ understanding acceptable wave functions. I am really trying to understand how to tell an unacceptable function. A question I have been given is the following:

Identify which of the following functions are acceptable wave functions:
a) plus minus x^2
b) cos theta
c) e^-ax where a is a constant

Wouldn't they all be not acceptable? Aren't a and c not finite everywhere (approach infinity with large x) and b have multiple values at certain points? Am I approaching this the right way? I don't know if I am completely wrong or if I am right by my reasoning isn't solid. Thank you for your help.

[clinz63]

Wave functions should be an eigenfunction of the Hamiltonian, right?

[Irlanur]

Wave functions should be an eigenfunction of the Hamiltonian, right?

They don't have to. More often than not in Quantum Chemsitry, they are not. (Note that a linear combination of Eigenfunctions is usually not an Eigenfunction).

To Bioinorganic78: If the question was asked exactly how you write it, they all are not, since the integral over the square does not converge. however, e.g. c) could be acceptable if it is only defined for x>0. Sometimes there are hidden assumptions like that...

[Enthalpy]

[...] Identify which of the following functions are acceptable wave functions:
a) plus minus x^2
b) cos theta
c) e^-ax where a is a constant
[...]

Hard to tell. It depends on if the extension of x or theta is limited, and on if the particle experiences a potential.

[Enthalpy]

Wave functions should be an eigenfunction of the Hamiltonian, right?

Have you possibly forgotten a detail? Say, what category of wave functions?

[Bioinorganic78]

We have just started quantum chemistry/mechanics, so I am not familiar with eigenvalues or Hamiltonian. However, the question was given exactly as I wrote, with no mention of permitted values or ranges.

If I am understanding this right, if a function is finite or square integrable, it simply means that is does not approach infinity for all values of x correct? No matter what value of x the function takes on, it will have a single, non-infinity value?