How are eigenvalues connected to the solutions of the Schrödinger equation?

[Leo32]

I'm reading an introductionary text on quantum physics and am stumbling a bit with the terms used.

The text discusses a finite potential box (one dimension, time independent). It calculates the conditions for the solutions of the wave functions, which I can follow perfectly.

At that point however, the text starts mentioning eigenvalues, completely out of the blue. After pondering already a few hours over what the autor might mean by it, the only possibility I see is that these are the eigenvalues of the Hamiltonian on the wave function.
From these eigenvalues, the text loops foreward to present the solutions right away.

Can somebody help me complete the jump from local conditions for wave functions, to eigenvalues, and then to actual solutions ?

Thanks !
Leo

 

[selfAdjoint]

The book should have a discussion of operators that operate on the wave function. The Hilbert operator is one of them, but there should also be position and momentum operators. Since they operate on the complex wave function they are complex operators. The book should state or show that these operators are Hermitian (or it may possibly say self adjoint , and that this makes the eigenvalues of those operators real. Then when you apply the operator to the wave function you get a set of real eigenvalues and eigenvectors of the wave function. The eigenvector represents a possible outcome of your observation and the eigenvalue, when normalized, gives the probability of that outcome among the set.

 

[dextercioby]

The book should state or show that these operators are Hermitian (or it may possibly say self adjoint , and that this makes the eigenvalues of those operators real.

Any good QM book should not discuss hermitean/symmetrical operators (hence nonselfadjoint) wrt physical observables.Though the distinction between those type of operators is merely mathematical subtility,it was important enough to be stated in the Second Principle of QM,where the notion of "self adjoint" is used to depict liniar operators (on the separable Hilbert space of states) associated by the so-called "quantization" to the obserable quantities.

 

[dextercioby]

I'm reading an introductionary text on quantum physics and am stumbling a bit with the terms used.

The text discusses a finite potential box (one dimension, time independent). It calculates the conditions for the solutions of the wave functions, which I can follow perfectly.

At that point however, the text starts mentioning eigenvalues, completely out of the blue. After pondering already a few hours over what the autor might mean by it, the only possibility I see is that these are the eigenvalues of the Hamiltonian on the wave function.
From these eigenvalues, the text loops foreward to present the solutions right away.

Can somebody help me complete the jump from local conditions for wave functions, to eigenvalues, and then to actual solutions ?

Thanks !
Leo

It's better if you change the book and consider something else to guide you through learning QM.A nice description of the problem appears in A.Messiah:"Quantum mechanics",volume 1,chapter 3 (the edition is irrelevant) and in Siegfried Fluegge"Practical Quantum Mechanics I",Springer-Verlag Berlin Heidelberg New York,1971,chapter 2 ("One body problems without spin").
Good luck!

 

[James R]

Leo32,

Sounds like you're on the right track. The time-indepedent Schrödinger equation is:

$$H\psi = E\psi$$

where H is the Hamiltonian and E is the energy.

This is an eigenvector equation, where the linearly-independent solutions are the eigenvectors of H, and the corresponding eigenvalues are the values of E which make the equation hold for the different $$\psi$$s.

 

[Leo32]

The book should have a discussion of operators that operate on the wave function. The Hilbert operator is one of them, but there should also be position and momentum operators. Since they operate on the complex wave function they are complex operators. The book should state or show that these operators are Hermitian (or it may possibly say self adjoint , and that this makes the eigenvalues of those operators real.

It does carry the discussion, but not in an "applied" way so to speak. I was just strugling to connect the more or less mathematical theory to the application in this case. Think I got it now however, but just to check:

- The Schrödinger equation is basically a differential equation, which also contains the conditions for the calculations of eigenvalues (Hamiltonean being the matrix, E being the eigenvalue and the wave function being the eigenvector)
- Calculate solutions for the Schrödinger equation for various conditions (here, the presence of a fixed potential over a certain range)
- Put foreward conditions that the wave functions are continuous and their derivatives as well
- Make sure the integral of the probability (psi*psi) of the total interval = 1

Thanks !

Leo