# Define quantum mechanics

1. Oct 19, 2011

### Demystifier

I would like to challenge you all to define quantum mechanics.
An ideal definition should contain only one sentence and should cover all aspect of quantum mechanics. If you cannot construct an ideal definition, you can try with a good but not ideal one which covers most but not all aspects of quantum mechanics, and/or contains more than one sentence.

Here is my try:

Definition 1 (probability amplitude):
Probability amplitude is a complex number the squared absolute value of which is equal to probability.

Definition 2 (quantum mechanics):
Quantum mechanics is a branch of physics which studies probabilities of different measurement outcomes, for systems for which the probability can be most easily calculated from a probability amplitude.

Last edited: Oct 19, 2011
2. Oct 19, 2011

### lightarrow

3. Oct 19, 2011

### ZealScience

Aren't both definitions some of the postulates of quantum mechanics?

4. Oct 19, 2011

### ash_bang

If I as a complete physics numpty cannot understand your definition, then what is the point in defining something ....

It would be nice, however to see a definition from the community.

If nothing else it would perhaps initiate the discussion to do so.

Ash!

5. Oct 19, 2011

### dextercioby

Hrvoje is part of community.

As for the sought definition, it's merely:

Quantum Mechanics is the theory whose predictions follow logically from the following set of axioms:

1.
2.
3.
4.
5.
...

Last edited: Oct 19, 2011
6. Oct 19, 2011

### Demystifier

It's certainly legitimate, but probably not ideal for a definition. By definition, a mean a concise statement that could be written e.g. in a science dictionary or at the beginning of a Preface in a quantum-mechanics textbook. As such, a definition should be precise but still not too technical. It's of course difficult to achieve, but that's precisely why this thread should be challenging.

7. Oct 19, 2011

### Demystifier

No. For example, even a probability in classical statistical mechanics can be expressed in terms of probability amplitudes, but that's not the easiest way to formulate classical statistical mechanics.

8. Oct 19, 2011

### jambaugh

OK, Here's my attempt...

• As with CM, QM associates every system observable with a corresponding generator (in some Lie algebra) of a group of transformations (potential symmetries of the system) a la Noether's theorem.

• As with CM, QM defines the dynamics of a system by the action of one of these generators, the Hamiltonian.

• Where in CM the most general system description is a a probability distribution on a manifold of states (state space typically phase space), in QM the system description is a linear functional defining expectation values on the Lie algebra of observables, satisfying appropriate consistency conditions. (e.g. $\langle 1\rangle = 1$ "the system exists" so we expect it to exist), and as such transforms dually to the observables under the action of the Hamiltonian or other generators of transformations in the system's Lie algebra.
• and finally, as with CM, in QM the act of incorporating knowledge of a specific observed value should lead to our modifying the system description so that immediate subsequent measurement is predicted to produce the same value with certainty.

I think this paradigm is sufficient to generate all of QM once one carries out a program of empirical experiments to see which Lie algebra corresponds to a given physical system. I don't mention representations of the Lie algebra because that is just an embedding into a larger Lie algebra.

Some Details:
We begin in both cases with a probabilistic description and then see what logical certainties may be represented. We can describe a classical system in a given state using a Dirac delta distribution over the state manifold.

The linear functional for QM system description is the mapping $A \mapsto \langle A \rangle = \mathop{Tr}(\rho A)$. The maximal description is when $\rho$ is proportional to a minimal non-zero projection operator in some representation of the Lie algebra in question.

9. Oct 19, 2011

### ZealScience

But in classical mechanics, they don't use vector space, right? I think phase space has nothing to do with complex vectors which you mentioned in your definition.

10. Oct 19, 2011

### lightarrow

QM is the theory in which fields' energy is not continuous but an integer number of hv.

11. Oct 19, 2011

### dx

It is a mathematical formalism that generalizes classical mechanics, adapted to describe physical phenomena in which h (Plank's constant) cannot be neglected.

12. Oct 20, 2011

### Demystifier

13. Oct 20, 2011

### Demystifier

It's true, but for many aspects of QM this fact is not really important.

14. Oct 20, 2011

### Demystifier

It's true, but it explains nothing. Someone who does not already know what QM is, after reading this will still have no idea what it is.
At least, one should add one additional definition, which defines the Planck constant.

15. Oct 20, 2011

### jambaugh

For Goka's definition yes. But given it is an empirically determined quantity one need not define its value. It should be sufficient to state or imply that some observables are quantized with one's definition.

To All: I think the "meta-question" should be hashed out here. What do we want in a definition?

• Should it be sufficient to reconstruct QM given sufficient empirical experimentation?
• Should it include foundational experimental results (e.g. Einstein's photo-electric effect).
• Should it be sufficient to allow an educated layman to understand QM? [tall order!]
• Should it be axiomatic? Operational? Minimalistic?
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To my thinking, one must imply or explicitly invoke:
• Born's probability formula or the equivalent,
• The Eigen-Value Principle or some equivalent expression of observable spectra,
• Hamiltonian dynamics.

I don't see these included without pulling out the heavy duty algebra which will get a bit esoteric for the layman.

I'd also suggest practicing with "Define classical mechanics" as a warm up.

16. Oct 20, 2011

### lightarrow

For example? (Just in order to clarify it to myself).

17. Oct 20, 2011

### Demystifier

For example, superposition and entanglement.

18. Oct 20, 2011

### Demystifier

That's a good question. I would say it should be minimalistic, but by that word I probably do not mean the same as you do.

19. Oct 20, 2011