Homework Statement
Consider two observables A and B such that [A,B]=0. Given the spectral resolutions of each operator:
A = \sum_k a_k P_k
B= \sum_j b_j Q_j
where P_k and Q_j are projectors onto the eigenstates of their respective operator.
Show that [P_k,Q_j]=0 for every k and jThe...
Does anyone know the answer to this:
Consider a rigged hilbert space \Phi \subseteq H \subseteq \Phi^*. I know that the nuclear spectral theorem says any member of \Phi can be expanded in the generalized eigenfunctions of the position operator. I was wondering if any member of \Phi^* can be...
Is the domain of every bounded operator on a Hilbert space the entire space? I think I remember reading that so I'm wondering if this is what you're talking about.
Also, a few small asides:
1) Many people in my theoretical chemistry department say a bounded operator is one which has bounded...
Thanks very much, that was really help helpful. I think what you're saying is that since the generalized position eigenstates lie outside the Hilbert space they cannot be considered a basis by definition . This means that the size of \{|X\rangle\} says nothing about the dimension of the subspace...
Hi,
I was wondering how the state vector for a particle in a 1-D box can be expanded as a linear combination of the discrete energy eigenkets as well as a linear combination of the continuous position eigenkets. It seems to me that this is a contradiction because one basis is countable whereas...
Homework Statement
What minimal speed do you need to throw a stone with so that it starts orbiting the Earth? What minimal speed do you need to throw a stone with so that it flies off to infinity (forgetting about the Sun)?Homework Equations
The motion of the stone takes place in an effective...
If a charged particle is moving in an electrostatic field is angular momentum conserved? I'm thinking it's only conserved if the electrostatic potential is constant throughout space
Hi, I am trying to solve this problem here:
http://img201.imageshack.us/img201/7006/springqo9.jpg
We're supposed to find the equation of motion from the lagrangian and not Newton's equations.
Attempted solution:
L = T - U = \frac{I\omega^2}{2} + \frac{mv^2}{2} - \frac{kr^2}{2}
I = m(r^2 +...
oh it was actually asking for the probabilities for the individual n=2 orbitals
i received probabilities of 0 for 2s and the 2p orbitals corresponding to the quantum numbers m = +/- 1. the only non zero probability transition i received was for 2p corresponding to m = 0.
the field is just a...
Homework Statement
A Hydrogen atom in its ground state (n,l,m) = (1,0,0) is placed in a weak electric fieldE(t) = 0 if t < 0
Eo *e^{\frac{-t}{\tau}} if t > 0E is in the positive z direction
What is the probability that it will be found in any of the n=2 states at time t > 0 ? use...
Homework Statement
Find the taylor series of \frac{1+z}{1-z} where z is a complex number and |z| < 1
Homework Equations
\sum^{\infty}_{0} z^n = \frac{1}{1-z} if |z| < 1
The Attempt at a Solution
\sum^{\infty}_{0} z^n = \frac{1}{1-z}
\frac{1+z}{1-z} =...
It is known that a particle in a one dimensional box with walls at (-a/2, +a/2) has energy probabilities:
P(E1) = 1/3, P(E2) = 1/3, P(E,3) = 1/3, P(En) = 0 for all other n. If the parity of the state is measured and +1 is found, what can you say about the value of the measurement of E sometime...
When you make ice cream using a couple of zip loc bags and a salt/ice mixture do the room and surroundings have any effect? Can you be in a warm room and still make the ice cream? Thanks.