Energy eigenstates White board activity White board activity x y
Ronck, Abigail, Features Editor has reference to this Academic Journal, PHwiki organized this Journal 3.1 Schrödinger Equation Postulate 6: Schrödinger Equation The time evolution of a quantum system is determined by the Hamiltonian or total energy operator H(t) through the Schrödinger equation = Operator corresponding to total energy Derived from classical Hamiltonian What kind of terms do we expect it could contain How is H different from E Lets see what this equation tells us about our states if H is time-independent is a functional: traditionally a map from a vector space of functions usually to real numbers. In other words, it is a function that takes functions as its argument in addition to returns a real number (or it can be written in matrix as long as m as an operator) As an operator corresponding to an observable, what properties do we expect it to have If we diagnonalize it, what will the eigenvalues correspond to Combine the above equation with the Schrodinger equation to determine what we can say about the eigenvectors (eigenstates) 3.1 Schrödinger Equation Postulate 6: Schrödinger Equation The time evolution of a quantum system is determined by the Hamiltonian or total energy operator H(t) through the Schrödinger equation = Operator representing total energy Argued Friday: If H is time-independent, in addition to we write this in the continuous functional as long as m: ih(d(t)/dt) = H(t) The time derivative returns itself, with an i out front So we can write (t) = eitn, where n is time-independent So in Dirac notation we may expect: (t)> = eitn> is a functional: a map from a vector space of functions to numbers. In other words, it is a function that takes functions as its argument in addition to returns a number often used in the calculus of variations to find the minimizing function here that would be finding the state that minimizes energy
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As an operator corresponding to an observable, we know its Hermitian, so it must contain a complete basis of eigenkets in addition to eigenvalues which correspond to energies of our system Here Ei> is our n>, weve taken out the time-dependence NOTE H is an operator, Ei are real, scalar, eigenvalues the Hamiltonian is NOT the energy! Lets put time back into the states if the full eigenstates (t)> had no time-dependence then d/dt would yield zero, but the time dependence doesnt live in the eigenstate of the hamiltonian since H has no time dependence. AND, the eigenvectors must satisfy the completeness relationship, so we must be able to write: Shove into the Schrödinger equation Use orthonormality to simplify Verify what our underst in addition to ing of continuous functions told us Simplify Get general as long as m of (t)> as long as all time-independent Hamiltonians: (t)> = iCoe-iwitEi> thats an wi in the exponential! (Yucky microsoft ) What will happen if our system starts in one particular energy eigenstate at t=0, say E2>, then we watch it as long as some time t how do we write the state at time t How will the probability of finding the state with energy E2 (or any energy) after time t differ from at time t=0 How will the probabilities associated with any observations of this state after time t differ from at t=0 What name might you give such an initial pure state to explain how it behaves with time
If our initial state at t=0 is aE1> + bE2>, what will be our state after some time t How will the probability of finding the state with energy E2 (or any energy) after time t differ from at time t=0 How will the probabilities associated with any observations of this state after time t differ from at t=0 Reminders: found that the energy eigenstates are stationary states, in addition to that if H is time independent, we can exp in addition to any state as: (t)> = iCie-iEit/Ei> Refresher calculation: our initial state at t=0 is aE1> + bE2>, what will be our state after some time t, in addition to what is the probability of finding the state with energy E2 (or any energy) at any time t Your turn: limit to 2-level system as long as simplicity How will the probabilities associated with any observations of this state after time t differ from at t=0 Energy eigenstates Regardless of our initial energy eigenstate, the probability of observing the energy of the state does not change with time (thats because our Hamiltonian, which represents total energy, is time independent!) (This is true as long as any observable that commutes with H as well since commuting observables share eigenstates) However, if we are considering a general observable, our probability will oscillate with time! angular frequency of the time evolution = Bohr Frequency 12 = E2-E1/
Measurement of some other observable A say position, momentum, or spin!! If [A, H ] = 0, then observations wont change with time stationary states since A in addition to H must share eigenstates (2) If [A, H ] 0, then observations will oscillate with time Look at example of time evolution of square well states using the PhET simulation (http://phet.colorado.edu) Explain what you observe in terms of our new underst in addition to ing of time evolution of quantum states Recipe as long as solving a st in addition to ard time-dependent quantum mechanics problem with a time-independent Hamiltonian Given Hamiltonian H in addition to an initial state y(0), what is the probability that an is measured at time t Diagonalize H (find eigenvalues Ei in addition to eigenvectors Ei). Write y(0) in terms of energy eigenstates Ei. 3.2 Spin Precession Hamiltonian of a spin-1/2 system in a uni as long as m magnetic field (The electron g-factor is a bit more than two, in addition to has been measured to twelve decimal places: 2.0023193043617) Find our measurements depend on energy differences, so our Hamiltonian needs to only include terms that will involve energy differences in the two possible spin states Only the dipole potential energy does this: recall that U = -·B Choice as long as zero point of potential energy is arbitrary if we only care about energy differences No kinetic term is needed here, K=0 below
3.2.1 Magnetic Field in z-direction The uni as long as m magnetic field is directed along the z-axis. This is a time independent hamiltonian!! We can just write down the eigenstates since they must be the same as as long as H since Sz in addition to H commute The probability as long as measuring the spin to be up along the z-axis The initial state is an energy eigenstate. The time evolved state simply has a phase factor in front. there as long as e there is no physical change of the state with time Example (1) The initial state is spin up along z-axis: + in addition to – are stationary states, if we start purely in one of them, we stay there!.
Example (2) The most general initial state: In matrix as long as malism This overall phase out front does not effect measurements Only this term changes it is rotated to a new angle phi, but theta has stayed the same! White board activity Find the probability finding the general state in spin up in z, in x, or in y (row 1, 2 in addition to 3): (t)> Time independent since the Sz eigenstates are also the energy eigenstates, in addition to are there as long as e stationary states. Consistent with the fact we found the polar angle q to be constant. Time dependent since the Sx eigenstates are not stationary states ([Sz,Sx] is not zero).
White board activity Find
x y f +w0 t q f Spin Precession q w0 t Larmor frequency: frequency of precession What is
: if we start in a +>x state If we start in a +>y state If we start in a +> state What does this mean
Classical expectations: Assume the magnetic moment is aligned with its angular momentum is the gyromagnetic ratio, proportional to q/2m Torque will be perpendicular to the moment in addition to to B! It will change the direction of the angular momentum just like precession of a top If >0, precession is clockwise. In the spin-1/2 case as long as electron, it is negative (spin in addition to magnetic moment are anti-parallel) so the precession is counterclockwise This precession shows our system has angular momentum!! 3.2.2 Magnetic Field in general direction This will not have the same eigenvalues in addition to eigenvectors as the last case! We must diagonalize it!! What does this become if B is along z, so Bx = 0
Now we simply get: Let n be the unit vector in the direction of the magnetic field: Now were using our field direction to define the coordinate system: This makes sense were shifting our perspective to a new axis z which has been rotated from z by an angle of theta We already know how to deal with this system, because we know the eigenstates corresponding to Sn: We want to see if it is possible as long as a system that starts in the state +> to end up in the state -> (or vice versa) in this magnetic field that is aligned along z in addition to x Lets start in +>: Probability oscillates with frequency dependent on delta(E)
Super-K (Kamiok in addition to e) http://en.wikipedia.org/wiki/Super-Kamiok in addition to e Why are neutrinos massive In St in addition to ard Model, fermions have mass because of interactions with Higgs field (Higgs boson), but this cant explain neutrino mass Mass of neutrino is at least 500,000 times smaller than the mass of an electron, so any correction to st in addition to ard model wouldnt explain why the mass is SO small This remains unexplained http://en.wikipedia.org/wiki/Neutrino-oscillations Extending our knowledge CabibboKobayashiMaskawa matrix specifies the mismatch of quantum states of quarks when they propagate freely in addition to when they take part in the weak interactions IF this matrix were diagonal, there would be no mixing Necessary as long as underst in addition to ing Charge Parity violation (important research topic) (Nobel prize as long as 2008) http://en.wikipedia.org/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa-matrix
Ronck, Abigail Features Editor
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