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Mar
31
comment Why do we need both Hamiltonian and Hilbert Space to specify a Quantum System?
An Hamiltonian in Quantum Mechanics is an operator on some Hilbert space, usually unbounded but self-adjoint. How can you have an Hamiltonian if you don't know where it acts? For example, suppose that $H$ is the operator that stretches every vector by the positive amount $\lambda$. How would you represent $H$ as a matrix? Is it 2x2, 3x3, 147x147?
Feb
22
comment Where does the formula for the moment of inertia come from?
It comes from the definition of angular momentum when you enforce the form $L = I\omega$
Jan
25
comment Kronecker sum or direct sum?
What's $H$? A Hilbert space? Some operator? Some more context would definitely help.
Jan
23
comment Why doesn't the 9th ball move in the break in the nine-ball pool game?
Newton's cradle can have as many bodies as one likes.
Jan
23
revised Why doesn't the 9th ball move in the break in the nine-ball pool game?
added 39 characters in body
Jan
23
comment Why doesn't the 9th ball move in the break in the nine-ball pool game?
Assume a hit on the line through the centres of the 1, 9 and 8 ball for simplicity. By the symmetry all actions and reactions on the 9 ball cancel out. Same principle as for the Newton's cradle.
Jan
18
comment Lagrangian mechanics - small oscillations around equilibrium diagonalization
Observe that $\det(K-\omega^2M) = \det(M)\det(M^{-1}K-\omega^2I)=\det(M)\det(M^{-\frac12}KM^{-\frac12}-\omega^2I)$‌​. Now $M^{-\frac12}KM^{-\frac12}$ is diagonalisable and $\det(M)\neq0$ by hypothesis. However $M^{-1}K$ is not guaranteed to be diagonalisable unless, e.g., $K$ and $M$ commute. Perhaps there are extra assumptions on the Lagrangian?
Jan
7
comment Why does the preservation of transition probabilities imply the preservation of all quantum probabilities?
for the function $x^2$ you have $(UOU^*)^2 = UO^2U^*$, hence $p(UOU^*)=Up(O)U^*$ for any polynomial $p$. By Stone-Weierstrass the same is then true for continuous functions. To get the characteristic function of a Borel set simply take pointwise limits of continuous functions.
Jan
7
comment Negative powers of operators
Observe that $R^{-1}$ might not exist...
Jan
7
comment Why does the preservation of transition probabilities imply the preservation of all quantum probabilities?
The action of the symmetry can be transposed on observables, and it turns out to be $O\mapsto\mathcal SO = UOU^*$. When you compute a transformed expectation value you are changing both the state and the observable according to $\psi\mapsto U\psi$ and $O\mapsto UOU^*$, whence the invariance.
Jan
5
answered If $z_G$ is a principal axis that goes through the center of mass, are every other axes $z$ parralel to $z_G$ also principal axes?
Jan
4
comment Gravitons, photons and conservation of momentum
The notion of a field is introduced to salvage the conservation of energy and momentum. The field acts as a reservoirs of both the quantities, which are mediated by the quanta of the field (on a quantum level).
Jan
3
answered Poincare group representation and complete set
Jan
2
comment If we consider the electric field to act upon charges with a force, how does it stay in line with Newton's laws?
In a conductor with zero resistance, all the energy acquired by the charges would remain in the charges. But you can think of resistance as a sort of friction that is stripping kinetic energy away from charges and turning it into heat. The drift velocity is just the equilibrium regime of this scenario.
Jan
1
comment How can we fix the constant of the energy eigenstates of a quantum free particle such that they satisfy the orthonormality condition?
You have basically answered your own question. The kind of normalisation you're looking for is not the "standard" one, but a $\delta$-normalisation.
Dec
28
answered Difficulty in understanding ket vectors in quantum mechanics
Dec
28
comment Contradictory argument to continous energy spectrum in quantum mechanics
Observe that the solutions to the free particle don't lie in $L^2$ but in the rigged Hilbert space.
Dec
24
comment Calculating moment of inertia for a cylinder?
how about $\text d m = 2\pi h\rho r\text dr$, to integrate on $r\in [0,R]$?
Dec
23
comment Two different formulas
I suppose that for the general formula $\mathbf v$ is assumed to be a vector field. For a point-like particle differentiation of its velocity doesn't make too much sense.
Nov
28
comment Expectation to uncertainty
The product $AB$ doesn't yield an observable in general, so I wonder if one can really interpret the OP's formula as a covariance. Perhaps one can consider the Jordan product $A\circ B=\frac12(AB+BA)$ instead and set $\operatorname{cov}(A,B) = \langle A\circ B\rangle - \langle A\rangle\langle B\rangle$?