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  1. B

    Commutator and hermitian operator problem

    What this example also shows is that in *finite* dimensions you cannot have two hermitian operators A and B satisfying [A,B]=cI
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    Quantum Energy Levels

    For j=1/2 there are two states: m=+/- 1/2, and for j=3/2 there are 4 different m-states. The "weak" Zeeman effect just refers to a situation where the energy shift due to the magnetic field is small and can be treated with perturbation theory: the unperturbed Hamiltonian has split the l=1 level...
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    Quantum Energy Levels

    It's because the interaction that splits the energies of the state is the spin-orbit coupling, proportional to \vec{L}\cdot\vec{S}, which can be rewritten as being proportional to the difference \vec{J}^2-\vec{L}^2-\vec{S}^2, which is dependent only on the quantum numbers j and l (s=1/2 in...
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    Uncertainty Principle

    Once you have answered part (b), you will see that the relation that you're supposed to prove in part (a) cannot be quite correct!
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    Proving a matrix is unitary

    You clearly did not state the full problem so I have to keep guessing: were you supposed to diagonalize the Hamiltonian and find U such that [tex] H=\sum_k E(k) b^+_kb_k [/itex]?
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    Proving a matrix is unitary

    If the b's are fermionic annihilation operators, then that *means* they satisfy the anticommutation relations that, as you figured out, are equivalent to U being unitary. Done.
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    Proving a matrix is unitary

    You need more information to prove any of those relations. You must have been given some info about what the b's are supposed to be, for instance. I assumed that you had been told that the b's are fermionic annihilation operators.
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    Electrostatics charge density

    First, is the vector \vec{r} in E meant to be the *unit* vector in the r-direction, or not?
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    Trapped Electrons

    It's because a photon carries angular momentum, but in 1-D the concept of angular momentum is not defined.
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    Print ViewSimultaneous Measurements of Position and Velocity

    So far so good! Now the other thing you have been given is that delta v_x is 1% of v_x.
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    Normalizing Wave Function of A Ring

    You just took the square of the wavefunction, instead of the absolute value squared.
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    Average net force

    There are (at least) two sorts of average you can take: If you want to calculate a force averaged over *distance*, you can use the change in kinetic energy, divided by distance: Work =\delta E=F_average * d If you want to calculate an average over *time*, then that would be given by the change...
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    Proving a matrix is unitary

    No, you have to prove U is unitary. Edit: you already seem to know that U being unitary is equivalent to the b's satisfying the same anticommutation relations as the c's. But that's all there is to it....
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    Uncertainty principle questions.

    Nature itself has decided that there is no state in which both position and momentum of a a particle are sharply defined.
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    Finding a force from a simple potential

    Since U only depends on r, that's NOT the mistake OP makes.
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    Finding a force from a simple potential

    Yes, your dU/dr is wrong. Maybe if you write r^{-12} you'll see what your mistake is.
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    Calculating work done on a truck by friction when a truck is pushing a car

    The title of your post asks a different question that your post itself: which one is the actual question? And, don't you have more info?
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    Calculating work done on a truck by friction when a truck is pushing a car

    Not correct: the 300N force is NOT the force acting on the truck.
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    What is the average force?

    Looks good to me, except for you not including the correct unit for force.
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    Proving a matrix is unitary

    You should use that the c operators satisfy the same anticommutation relations that the b's also satisfy. On the other hand, c_p and b_q do not, in general, satisfy such relations.
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    1D simple harmonic oscillator in box

    Oops: didn't notice the infinite! I misread this for a different problem where the outside potential is zero, somehow.
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    Wave Function Spherical Coordinates Probabilities

    <\Psi | L_z | \Psi> is the expectation (i.e., average) value of L_z. The probabilities to find specific values m for L_z follow from writing your wavefunction in the form |\Psi>=\sum_m c_m |l,m> as in your special case you only have 1 value of l in the superposition. In this case the...
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    Wave Function Spherical Coordinates Probabilities

    The one you displayed under "2. Homework Equations "!
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    Newton's first law

    Yes, there are in fact two action-reaction force pairs, one between Earth and car (gravity), the other between road and car (contact).
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    Newton's first law

    1. yes, with a small proviso: the gas that leaves the exhaust pipe would be considered as leaving the car, and thus contributing a tiny force in the forward direction! 2. yes, apart from the tiny effect mentioned above, the only external forces on the car in the horizontal direction are...
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    Newton's first law

    The engine plus gasoline plus tires plus dashboard plus windows etc. etc. etc. are all part of "the car". All forces that act internally to the car are parts of action-reaction forces and therefore do not give rise to a net external force. The motion of the car as a whole (more precisely, its...