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2

Hermitian operators (or more correctly in the infinite dimensional case, self-adjoint operators) are not used because measurements must use real numbers, but rather because we almost always decide to use real numbers. As the OP mentions at one point, you might choose to use complex numbers to label a two-dimensional screen, and in that case you'll be able ...


3

When in doubt go back to the masters. From Dirac's Principles of QM When we make an Observation we measure some dynamical variable. It is obvious physically that the result of such a measurement must always be a real number, so we should expect that any dynamical. variable that we can measure must be a real dynamical variable. One might think ...


3

In the Heisenberg picture, one simply has $$ A(t) = \exp(-Ht/i\hbar) A(0) \exp(+Ht/i\hbar) $$ The Hamiltonian $$ H = \frac{E_1+E_2}{2}\cdot {\bf 1} + \frac{E_1-E_2}{2} \cdot \sigma_z $$ while $$ A(0) = a\sigma_x $$ The term in $H$ proportinal to ${\bf 1}$ cancels in $A(t)$ so we have $$ A(t) = a\cdot \exp(-(E_1-E_2)t\sigma_z/2i\hbar) \sigma_x \exp(+(E_1-E_2)...


1

I have been wondering the same question recently. There is a way to define the isobaric partition function that seems to work pretty well with large systems, but I'm not convinced that it's appropriate with small ones. The idea is not to define a measure of the volume; instead define volume indirectly via the canonical ensemble, as follows. In the canonical ...


1

What you are correctly pointing out is that the calibrated scale attached to a measuring arrangement is arbitrary insomuch as it doesn't change the nature of the thing being measured. The thing is, conventional Quantum Mechanics identifies observables with self-adjoint operators (or equivalently their associated resolutions of the identity) and this object ...


5

Whilst it is certainly true that Quantum Probability Theory (QPT) is an entirely different framework from Classical (Kolmogorovian) Probability Theory (CPT) (specifically because the event structure is non-Boolean and the random-variable structure is non-commutative), we can still identify enough formal similarity to borrow the classical terminology. In ...


0

Maybe you can be interested in another interpretation of Hermitian matrices. In a recent paper we have proposed to see them as gambles on a quantum experiment. We have then enforced rational behaviour in the way a subject accepts/rejects these gambles by introducing few simple rules. These rules yield, in the classical case, the Bayesian theory of ...


1

A quantum system can be described by a set of evolving quantum mechanical observables. This is not the same as describing a system in terms of a stochastic quantity described by a single number chosen at random. A quantum system really does have multiple values of any unsharp observable, see https://arxiv.org/abs/quant-ph/0104033. Those different versions ...


7

I'm going to try to explain why and how density operators in quantum mechanics correspond to random variables in classical probability theory, something none of the other answers have even tried to do. Let's work in a two-dimensional quantum space. We'll use standard physics bra-ket notation. A quantum state is a column vector in this space, and we'll ...


6

I believe it is misguided to think that classical probability makes sense any more than quantum mechanics, with its "peculiar" probability calculations, makes sense. I'm going to be slightly mischievous here and make a friendly attack your first paragraph: does really make sense? Of course it makes perfect sense as a measure-theoretic definition, but how ...


11

Quantum mechanics is indeed a probability theory, but it is a non-commutative probability theory. So it is not just a matter of having signed/complex measures, but really of having a non-commutative probabilistic framework. Quantum mechanics was developed, historically, before non-commutative probability theories and I think that people in probability ...


9

You could certainly model any one quantum observable as a random variable. The problem comes in when you have multiple observables, which you might attempt to model as classical random variables with some joint distribution. From this joint distribution, you can compute various probabilities (like $\textrm{Prob}(Y\neq X)$, for example), according to the ...



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