I was reading John Gribbin’s “In Search of Schhrodinger’s Cat” last night, and got to thinking: I’ve always seen equations of the form delta x * delta p > h-bar or delta E * delta t > h-bar, but are those the only two such reationships? Do any two quantities that multiply together to produce units of g*m^2/s (what would that even be, an action?) have such a relationship? Are any other of these relationships meaningful in terms of commonly understood quantities?
-b
So this is saying that conjugate variables in quantum mechanics have an uncertainty associated with them. I suppose any other conjugate pairs would be similar.
What are the other conjugate variables?
Precisely what I’m asking. Energy and time are related by the uncertainty principle, as are position and momentum. Both of these pairs multiply together to yield units of (mass)*(distance)^2/(time), which, i suppose, are the units h-bar is in. Are there any other pairs of commonly understood quantities that, when muliplied together, yield such units, and if so, do they have the same uncertainty relationship? For instance, if one took density (g/m^3), the corresponding variable would be m^5/s. This is hardly a commonly encountered quantity, but would such a beast have an uncertainty relationship with density?
-b
One other set of complimentary quantities are the number of particles and the phase of the wavefunction. You can either count particles precisely or measure the wave preceisely, not both. The equation is Delta N * Delta Phi > 2 Pi. There’s no Planck’s constant involved.
Warning: overly technical explanation follows!
Mathematically, two quantities form complimentary pair if their operators are non-commutative. This means you get a different answer if you measure momentum and then position, or position and then momentum. For commutative operators, you’ll get the same answer no matter which order you measure–spin and then position, or position and then spin, will both get you the same results.
The Planck’s constant in the famous Heisenberg equation comes from the exact form of non-commutativity. Different pairs may have different constants.
A slight clarification to Pleonast’s post. There’s a general theorem that gives any uncertainty principle for two operators/observables A and B in terms of their commutator AB-BA. This is non-zero for non-commutative pairs and zero for commutative ones. It doesn’t involve Planck’s Constant. For observables that correspond to classical quantities, Dirac famously realised that the commutator is essentially equal to Planck’s Constant times a classical expression known as the Poisson bracket. The latter had even been introduced in classical mechanics long before. You can easily find an infinite number of pairs of classical quantities with non-zero Poisson brackets and hence which have quantum uncertainty relations. These tend to involve Planck’s Constant times a number, just like the simple examples.
Pleonast’s example is slightly different in that the phase of a wavefunction is an entirely quantum quantity and so the Dirac step doesn’t apply. That’s why it doesn’t involve h; there’s no classical equivalent.
Although there are infinitely many uncertainty relations, aside from the ones already mentioned, the only other ones that really crop up are those involving electromagnetic fields. These are the subject of a famous (but, in my experience, obscure) pair of papers from 1933 and 1950 by Bohr and Leon Rosenfeld. In his biography of Bohr, Pais quotes an anonomous friend about the first one: “It is a very good paper that one does not have to read. You just just have to know it exists.”
Don’t just tease us. What are they?
Well, if you’re going to insist… For measuring both electric and magnetic field strengths, in spatial region dL and in time dt, you get formulae like:
|dE||dH| > h/c(dtdL)(dtdL).
I’ve cribbed this from Pauli’s General Principles of Quantum Mechanics (Springer-Verlag, 1980, section 25c); I recall there’s a similar discussion in Gottfried and Weisskopf’s Concepts of Particle Physics (Oxford, 1984), but I’ve only got volume one to hand and it’s in the other one. As already sort of noted, Pais does a qualitative run round the topic in Niels Bohr’s Times (Oxford, 1991, section 16d) and that’s the nearest I know to a popular level explanation of it.