10 questions on QM postulates



Hello,

I am an undergrad physics student who just completed a course in modern physics
(mostly an introduction to QM). I have several questions related to the
postulates of Quantum mechanics which I find intriguing and which my textbook
imo did not address adequately. I have listed my questions below. I know there
are a lot of questions; answering just one at a time would be great.


1) How can the mathematical concept of hermitian operators be translated into
physical terms? What are all the conditions which ultimately restrict operators
such that they can be only hermitian as opposed to any other type?

2) how is it possible (or: how come it does not contradict the principles of QM)
that a continuum of eigenstates (as opposed to quantized eigenstates) can exist
for unbound states?

3) How can it be shown that the gradient (of psi when operating) times ?ihbar
(i.e. the momentum operator) is equivalent to h/lambda? Can any equivalence
between the momentum operator and the classical mechanical definition of
momentum (mv)be shown at all ( I know, it's called the wave equn, but still)?

4)How can it be shown directly (that is, without reverting to KE= p^2 /2m) that
the kinetic energy operator actually is a kinetic energy term?

5)How can it be shown directly (again, without reverting to H=p^2 /2m+ PE) that
the the time derivative (of psi when used in operation) times ihbar is actually
equivalent to the hamiltonian? I am asking these
questions because I find it intriguing that the time derivative of psi should
be proportional to its second spatial derivative (assuming constant potentials).
This does not strike me as something obvious, even though playing around with
the equations suggests something like this (which I still do not conceptually
understand):

Assume we can set H=-W (Can we really do this? Why or why not?)

then: -W= int(F.dx)=int (dp/dt dx)
(here, F and dx would obviously have to point in opposite directions for the
work to be negative)
finally, substitute the momentum operator, and we magically recover the left
side of Schroedinger's equation.
(BTW I have never seen a "force operator". I imagine that's because it is
closely linked with particles rather than waves. But is that a strong enough
reason to probhibit its use? Why?)

This brings me also to my next question:

6) What is the reason for distinguishing in the Schroedinger equation between
the Hamiltonian and total Energy? Is this related to the view that Planck's
constant can be considered to be the fundamental unit of action? Under what
circumstances is it valid to substitute total energy for the Hamiltonian?

7)What examples, other than the simple harmonic oscillator, can be used to
illustrate the correspondence principle, that is, the idea that in the limit of
large scales quantum mechanics reduces to classical mechanics? In particular,
how can we get from the Schroedinger equation to the classical Hamiltonian (or
vice versa)?

8) Is there any experimental evidence which would answer the following question:
does the probability of finding a particle in a volume element (psi*psi) change
if the shape of the volume element is changed (but the volume is kept the same
as before)? Have any experiments been conducted which examine the predicted and
actual probabilities of finding particles in volume-elements that are not
cubical?

9)Does psi have a dimension (like length?) Can psi itself ever end up being
orthogonal to the imaginary axis (i.e. be real, rather than complex)? If so,
under what circumstances? Would that have any effect on what we observe?
I ask this question because if its phase shift ever caused psi to lie exactly on
the real axis, this would seem to introduce a dimensional problem when we square
it: A complex number with some dimension multiplied by its complex conjugate
gives a real number with same dimension. A real number with some dimension
squared obviously gives a number with the square of the dimension.

10) Some versions of the postulates also mention this one:

The total wavefunction must be antisymmetric with respect to the interchange of
all coordinates of one fermion with those of another. Electronic spin must be
included in this set of coordinates

I understand that this postulate leads to the Pauli exclusion principle, but I
don?t know how it does so (i.e. how do you derive the pauli exclusion principle
from this postulate?). Also, the postulate itself is not clear to me, can
someone explain this to me?.


Again, I know this is a lot, so answering just a part at a time would be fine.
Thanks,

Armin

.

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