On Saturday, March 29, 2025 at 4:10:44 PM UTC-6 Brent Meeker wrote:
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 googled conjugate operators and this is what I got. No mention of
Fourier transform.*
In the context of linear algebra and operator theory, the "conjugate" of an
operator, often referred to as the Hermitian conjugate or adjoint, is obtained
by taking the complex conjugate of each element in the operator's matrix
representation and then transposing the matrix.
*AG*
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\] .
*What is the domain and range of the operator you call "Time per unit
change of a variable"? AG*
Brent
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To view this discussion visit
https://groups.google.com/d/msgid/everything-list/7fea5d78-3bda-4988-b093-9c3e6adeed5en%40googlegroups.com.
