How are the formulas in QM derived?

[arzie2000]

I'm not quite familiar in QM. All I can see are formulas, I'm familiar with diff.calc., integral, & diff.equations.

Do most QM formulas use these methods in deriving their formulas? if not, what else?

 

[GDogg]

Well, linear algebra for one. Then calculus. And Fourier analysis. And usually solving the Schrödinger equation means solving a differential equation.

 

[malawi_glenn]

well arriving at the fundamental expression, you first observe nature and make postulates. Then of course you take advantage of the mathematical formalism available.

 

[Loren Booda]

I recommend the straightforward experimental-theoretical derivation of section 5-2, "Plausibility Argument Leading to Schroedinger's Equation" (p. 128-134) in Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, 2nd edition by Robert Eisberg and Robert Resnick.

It is based upon

1. the de Broglie and Einstein postulates

2. conservation of energy

3. linearization

4. and (initially) the free particle, constant potential, sinusoidal solutions

 

[Demystifier]

I'm not quite familiar in QM. All I can see are formulas, I'm familiar with diff.calc., integral, & diff.equations.

Do most QM formulas use these methods in deriving their formulas?

Yes. Not all, but most.

 

[quetzalcoatl9]

the best thing you could do in this area would be to read the seminal works by Heisenberg, Schrodinger and Dirac. Once folks such as Wigner, von Neumann, Born, Jordan, Teller, etc. put QM on a solid mathematical footing (i.e. via operator theory), people lost interest in how the original equations were formulated.

 

[Billygoat]

Roughly QM was derived from classical mechanics along this path (Very roughly...)

Lagrangian CM --> Hamiltonian CM --> Poisson Brackets (Still CM) --> Commutator operators (Pretty much Quantum) --> Either Heisenberg or Schroedinger approach. There's much abstract math at each step of this 'outline', but studying any of these stages will help, maybe. (Remember that this stuff was developed over two or three decades by some pretty sharp people.)

 

[dextercioby]

Once QM was put firmly on axiomatical basis in any of its multiple formulations, all formulas are derived from the ones occurring in the axioms.

 

[meopemuk]

Roughly QM was derived from classical mechanics along this path (Very roughly...)

Lagrangian CM --> Hamiltonian CM --> Poisson Brackets (Still CM) --> Commutator operators (Pretty much Quantum) --> Either Heisenberg or Schroedinger approach.

Yes, this is how QM was developed historically, and this is how it is presented in many textbooks. However, this path strikes me as being somewhat illogical: we shouldn't be deriving a more general and exact theory (quantum mechanics) from its crude approximation (classical mechanics).

Fortunately, there is a more logical path, which doesn't take classical mechanics as its point of departure. The basic formalism of QM (Hilbert spaces, Hermitian operators, etc.) can be derived from simple and transparent axioms of "quantum logic"

G. Birkhoff, J. von Neumann, "The logic of quantum mechanics", Ann. Math. 37 (1936), 823

G. W. Mackey, "The mathematical foundations of quantum mechanics",
(W. A. Benjamin, New York, 1963), see esp. Section 2-2

C. Piron, "Foundations of Quantum Physics", (W. A. Benjamin, Reading, 1976)

Time dynamics and other forms of inertial transformations are introduced via representation theory for the Poincare group

E. P. Wigner, "On unitary representations of the inhomogeneous Lorentz group", Ann. Math. 40 (1939), 149

P. A. M. Dirac, "Forms of relativistic dynamics", Rev. Mod. Phys. 21 (1949), 392.

I listed just the most significant references. There are many more works along these lines. However, for some reason, these ideas have not percolated to the textbook level (at least not to the extent they deserve).

Eugene.

 

[quetzalcoatl9]

There are many more works along these lines. However, for some reason, these ideas have not percolated to the textbook level (at least not to the extent they deserve).

i believe that is starting to change, j.j. sakurai's text does a nice job as does Atkins (perhaps not super rigorous, but starts axiomatically via operator theory)

 

[meopemuk]

i believe that is starting to change, j.j. sakurai's text does a nice job as does Atkins (perhaps not super rigorous, but starts axiomatically via operator theory)

The best textbook that follows the Poincare-Wigner-Dirac approach to dynamics is S. Weinberg "The quantum theory of fields", vol.1. Highly recommended!

Eugene.