On 3/29/2025 8:40 AM, Alan Grayson wrote:
On Monday, March 17, 2025 at 9:45:50 PM UTC-6 Brent Meeker wrote:
On 3/16/2025 1:51 PM, Alan Grayson wrote:
1) What necessitates the use of complex numbers (whereas in GR
only real numbers are used)?
QM exhibits interference so it must have wave-like phases that can
add and subtract. It predicts probabilities which must be
positive numbers. So one way to do this mathematically is to have
probability /amplitudes/, Psi, that are the "square root" of
probabilities, Psi*Psi (where * denotes the Hermitian conjugate),
that have phases so they can interfere. Then the dynamics are
linear in the Psi.
2) What necessitates the postulates that some, but presumably not
all operators are non commuting?
3) With respect to 2), why is the non commuting difference i*h
(or i*hbar)?
It is conjugate pairs that fail to commute. See attached.
Brent
That's the definition of conjugate pairs, that they don't commute.
No it's not. They are variables related by a Fourier transform.
I notice that E and t are also considered conjugate pairs, but since t
is a parameter in QM and not an operator, how can that be
intelligible? AG
I've posted it before. In quantum mechanics /energy and the time per
unit change of a variable/ are conjugate variables. So they satisfy an
Heisenberg uncertainty relation, often written [math]\Delta E \Delta t
\geq \hbar[math\] . This is sloppy though and not quite right. What is
right is given any operator A and the Hamiltonian H defining the time
evolution of A, then [math]\Delta A \Delta H \geq \frac{1}{2} \hbar
[d<A>/dt][math\] .
Brent
Brent
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