Understanding Matrix Mechanics in Quantum Mechanics

[Jamister]

TL;DR Summary

What is matrix mechanics? Is it the same meaning as Heisenberg picture?

In a course of QM they mention Matrix mechanics. But what is it exactly? Is it just Heisenberg picture?

 

[PeroK]

Summary:: What is matrix mechanics? Is it the same meaning as Heisenberg picture?

In a course of QM they mention Matrix mechanics. But what is it exactly? Is it just Heisenberg picture?

It's not the Heisenberg picture. Heisenberg developed an alternative formalism for QM based on matrices. This was superseded by the more popular wave mechanics (i.e. solving the Schroedinger differential equation). In short, it is technically a lot easier to use the SDE.

See:

https://en.wikipedia.org/wiki/Matrix_mechanics

Both are equivalent and special cases of the more abstract Dirac formalism.

 

[Jamister]

It's not the Heisenberg picture. Heisenberg developed an alternative formalism for QM based on matrices. This was superseded by the more popular wave mechanics (i.e. solving the Schroedinger differential equation). In short, it is technically a lot easier to use the SDE.

See:

https://en.wikipedia.org/wiki/Matrix_mechanics

Both are equivalent and special cases of the more abstract Dirac formalism.

I tried to read from wikipedia but it's really not clear, it's more like historical review. do you know a better source? thank you

 

[PeroK]

I tried to read from wikipedia but it's really not clear, it's more like historical review. do you know a better source? thank you

I don't, I'm afraid. I don't know if anybody taught matrix mechanics after 1927 or so.

 

[hilbert2]

It seems to me that the matrix mechanics only attempted to solve the possible measured values (eigenvalues) of different observables, without going to the prediction of the trajectory produced by some initial condition like in time-dependent Schrödinger equation. Another problem is that the operators in an infinite-dimensional Hilbert space can't really be seen as matrices, especially if the dimensionality is uncountable.

 

[vanhees71]

Heisenberg's approach was intuitive first. He thought about, what's really observable concerning an atom (particularly the hydrogen atom which was the only one which seemed to work with the plain Bohr-Sommerfeld quantization), and he came to the conclusion that it's the spectral lines, i.e., the transition rates between energy levels ("orbits").

This brought him to develop a formalism for the the harmonic oscillator first, leading him to a quantum mechanical "reinterpretation of classical observables". He also developed an algebra for his scheme, and Born immediately realized that this (non-commutative) algebra is just the algebra matrices obey, but that one needs an infinite-dimensional matrix.

It's of course natural that this version of quantum mechanics, developed by Born, Jordan, and Heisenberg quickly after Heisenberg's heuristic breakthrough, was formulated in what we nowadays call the Heisenberg picture of time evolution, i.e., with the full time dependence on the operators (matrices in their matrix-mechanics formulation).

The hydrogen problem was solved by Pauli within matrix mechanics.

Of course, everything became a lot more easy to handle with Schrödinger's wave mechanics, and after a big fight between him and Heisenberg, which theory might be the correct one, Schrödinger proved the complete mathematical equivalence. The most elegant formulate, of course, was Dirac's general "representation free" formalism, showing that everything can be formulated on an abstract Hilbert space and operators acting on it.

 

[romsofia]

Weinberg discusses this in his book "Lectures on Quantum mechanics", in which he discusses the history+math+reasons. You can preview most of the chapter here, but I would suggest buying it if you can, one of the better graduate level quantum books: https://www.amazon.com/dp/1107111668/?tag=pfamazon01-20

Also, he is one of the few profs I've had that takes the time to discuss the idea as well (we spent 2 lectures on matrix mechanics!). He really enjoys the history of physics!

 

[Demystifier]

In a course of QM they mention Matrix mechanics. But what is it exactly?

It's just QM expressed in the language of matrices. For instance, instead of saying that momentum is an operator, in matrix meachanics one says that momentum is a collection of all matrix elements $\langle n|p|m \rangle$.

 

[vanhees71]

It's using matrix elements of the operators wrt. a discrete complete basis. For the Heisenberg algebra (i.e., with position and momentum operators as basic operators generating the observable algebra for spin-0 particles) you can use the energy-eigenbasis of the harmonic oscillator.

Heisenberg's original Helgoland paper is pretty incomprehensible, while the following papers by Born and Jordan and by Born, Jordan, and Heisenberg are very clear. There's also a textbook (1930) by Born and Jordan (in German).