Question about the best method to use for finding wavefunctions and eigenvalues

[rwooduk]

We have been covering the annhilation and creation operators in class.

You can use the annihilation operator to find the groundstate wavefunction, and then use the hamiltonian in terms of annhiliation and creation operators to find the energy eigen value of that state. (or you could put the wavefunction into the Schrödinger equation)

By acting the creation operatior on the groundstate wavefunction you can find the next excited state wavefunction.

My question is if a question asks you to find the eigenstates and eigenvalues for a particular system is it best to use these annihilation / creation operators or would it be better to use the Schrödinger equation in its 'original' form?

I think the benefit of the annihilation / creation operator method is that it allows you to find the quantised amount that each new energy level would be raised by, but you could also do this using the Schrödinger equation if you find the ground state energy and the next excited state energy.

I'm trying to get an idea of the methods used more generally at an advanced level.

Thanks in advance for any input!

 

[The_Duck]

Unfortunately the method of creation and annihilation operators only works for the very simplest quantum mechanical systems. When the Hamiltonian gets more complicated it can't be factored simply into creation and annihilation operators.

In a typical quantum mechanics class you'll learn about some other techniques for finding the energy eigenstates. In some simple cases, such as a finite square well, you can attack the Schrödinger equation directly. The hydrogen atom can by solved by a direct attack on the differential equation, or, if you are a bit clever, with creation and annihilation operators. But in most cases there's simply no known exact analytic solution to the Schrödinger equation and you have to resort to approximation techniques such as perturbation theory or variational methods.

Perturbation theory is the most important technique to understand to prepare for more advanced material such as quantum field theory.

 

[rwooduk]

Unfortunately the method of creation and annihilation operators only works for the very simplest quantum mechanical systems. When the Hamiltonian gets more complicated it can't be factored simply into creation and annihilation operators.

In a typical quantum mechanics class you'll learn about some other techniques for finding the energy eigenstates. In some simple cases, such as a finite square well, you can attack the Schrödinger equation directly. The hydrogen atom can by solved by a direct attack on the differential equation, or, if you are a bit clever, with creation and annihilation operators. But in most cases there's simply no known exact analytic solution to the Schrödinger equation and you have to resort to approximation techniques such as perturbation theory or variational methods.

Perturbation theory is the most important technique to understand to prepare for more advanced material such as quantum field theory.

Ahhh yes i wondered about its application to other quantum systems! We have only applied it to the simple harmonic oscillator, I was going to try and apply it to an infinite / finite squre well, are you saying its not possible using this method? May attempt the Hydrogen atom but really just trying to get an idea of what it works for and then choose the simplest method to use.

We have just started perturbation theory.

Many thanks for your answer! It's appreciated!

 

[moriheru]

The SE for hydrogen atoms is a bit more complicated to solve as you will see(at a cirten point it involves power series of the differential). You have to different equations actually for the small and the large...(sorry I was talking about the wavefunction)

 

[strangerep]

May attempt the Hydrogen atom but really just trying to get an idea of what it works for and then choose the simplest method to use.

To find useful c/a operators (or more generally, "ladder operators"), one must understand the Lie group theory applicable to each particular problem.

The basic harmonic oscillator is easiest (Weyl/Heisenberg algebra).

For quantum angular momentum, it's also (relatively straightforward, cf. Ballentine ch 7).

For the H-atom, one can obtain the energy spectrum by Lie-algebraic analysis of the (suitably quantized) conserved quantities of the Kepler problem, but that's quite nontrivial. (Pauli did it before the Schrödinger equation appeared, but that just proves he was a f--ing genius.) oo)

Bottom line: you've got to understand the dynamical symmetries of the problem before trying to find useful c/a operators. Usually, they don't just fall out in any obvious way. There's been some recent work by Odake, Sasaki and others on doing this for a wider class of potentials, but you'll need strong stomach to wade through their papers.

 

[dextercioby]

[...]

For the H-atom, one can obtain the energy spectrum by Lie-algebraic analysis of the (suitably quantized) conserved quantities of the Kepler problem, but that's quite nontrivial. (Pauli did it before the Schrödinger equation appeared, but that just proves he was a f--ing genius.) oo)

To be fair (to my knowledge!), up to 1966, only the discrete spectrum of the Hamiltonian (of the 'dummy particle' in the Schrödinger eqn.for the H-atom) was 'solved' algebraically, in particular by Pauli in 1925, then of course by V.Fock in 1935 (<Hydrogen Atom and Non-Euclidean Geometry>,Zs. Phys., 98, N 3-4, 145, 1935). Oh yeah, Pauli was a very arrogant genius. :D