Orthonormal wavefunctions?

In summary, to prove that two wavefunctions with respect to (r, theta, phi) are orthonormal, one would let the first wavefunction be denoted as Y(psi) and the second as Y* (psi star). Then, one would integrate the product of the first and second wavefunctions over the appropriate bounds, using the measure for spherical polar coordinates. This would result in a volume integral, with the functions being multiplied by the appropriate factors for r, theta, and phi. This method is commonly used to prove orthonormality in spherical polar coordinates.
  • #1
MontavonM
7
0
Given 2 wavefunctions with respect to (r,theta,phi)...

To prove that the functions are orthonormal, you would let the first w.function = Y(psi) and the 2nd = Y* (psi star), then you would integrate --> SY*Y dr dtheta dphi (integral of psi* times psi) dr dtheta dphi.

Correct?
 
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  • #2
yeah, over a whole period, or the appropriate bounds.
 
  • #3
MontavonM said:
Given 2 wavefunctions with respect to (r,theta,phi)...

To prove that the functions are orthonormal, you would let the first w.function = Y(psi) and the 2nd = Y* (psi star), then you would integrate --> SY*Y dr dtheta dphi (integral of psi* times psi) dr dtheta dphi.

Correct?

That looks like the wrong measure for spherical polar coordinates.
I would have expected something like:
[tex]
\int_0^{2\pi} d\phi \int_0^\pi \sin\theta \; d\theta \int_0^\infty \bar f g \, r^2 dr
[/tex]
where f,g are functions of [itex]r,\theta,\phi[/itex].

See also:

http://en.wikipedia.org/wiki/Volume_integral
 

1. What is the definition of an orthonormal wavefunction?

An orthonormal wavefunction is a mathematical function used to describe the quantum state of a particle. It is characterized by being normalized (having a magnitude of 1) and being orthogonal (perpendicular) to other wavefunctions in the same system.

2. How are orthonormal wavefunctions used in quantum mechanics?

Orthonormal wavefunctions are the basis for calculating the probability of finding a particle in a certain state. They are also used to calculate the time evolution of a quantum system and to determine the energy levels of a system.

3. What are the properties of orthonormal wavefunctions?

Orthonormal wavefunctions have a magnitude of 1, are orthogonal to each other, and are complex-valued. They are also eigenfunctions of the Hamiltonian operator, meaning they represent solutions to the Schrödinger equation.

4. How do you determine if two wavefunctions are orthonormal?

To determine if two wavefunctions are orthonormal, you must calculate their inner product (also known as the overlap integral). If the result is 0, then the two wavefunctions are orthogonal and if the result is 1, then they are orthonormal.

5. What is the significance of orthonormal wavefunctions in quantum mechanics?

Orthonormal wavefunctions are essential in quantum mechanics as they allow us to calculate the probabilities of different states of a system. They also provide a basis for expanding the wavefunction of a system, making it easier to solve complex problems.

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