GQM:010117Cyclic e1 on the Jacobi matrix [[0,1,0],[1,0,1],[0,1,0]] has weights 1/4,1/2,1/4 and G11(3)=8/21Graduate Quantum Mechanics 1 Jacobi Spectral WeightsGraduate Quantum Mechanics 1 Stieltjes Moments CyclicPrivateLabels unavailablePrivate saving unavailableHilbert spaces, unbounded operators, and spectral structureGraduate Quantum Mechanics I · Mixed reviewProgress not loaded
GQM:010118Riesz projectors of A=[[1,1],[0,2]] are P1=[[1,-1],[0,0]], P2=[[0,1],[0,1]] and exp((ln2)A)=[[2,2],[0,4]]Graduate Quantum Mechanics 1 Nonnormal Riesz ProjectorsGraduate Quantum Mechanics 1 Holomorphic Functional CalculusPrivateLabels unavailablePrivate saving unavailableHilbert spaces, unbounded operators, and spectral structureGraduate Quantum Mechanics I · Mixed reviewProgress not loaded
GQM:010119H=diag(-1,0,1)+2vv^dagger with v=(1,0,1)/sqrt(2) has spectrum 1-sqrt(2),0,1+sqrt(2) and v-weights 1/2∓1/(2sqrt(2))Graduate Quantum Mechanics 1 Rank One InterlacingGraduate Quantum Mechanics 1 Spectral Weights UpdatePrivateLabels unavailablePrivate saving unavailableHilbert spaces, unbounded operators, and spectral structureGraduate Quantum Mechanics I · Mixed reviewProgress not loaded
GQM:010220On Schwartz space the Weyl pair T(pi/2),B(1) has group-commutator phase -i and TBpsi=e^{i(x-pi/2)}psi(x-pi/2)Graduate Quantum Mechanics 1 Weyl Ccr PhaseGraduate Quantum Mechanics 1 Translation Boost ActionPrivateLabels unavailablePrivate saving unavailableQuantum symmetries and continuous generatorsGraduate Quantum Mechanics I · Mixed reviewProgress not loaded
GQM:010221In hbar=1 units, dilation of the Gaussian at s=ln2 yields psi_s=2^{-1/2}pi^{-1/4}e^{-x^2/8} with VarX=2, VarP=1/8, product 1/4Graduate Quantum Mechanics 1 Dilation Generator CommutatorsGraduate Quantum Mechanics 1 Gaussian Dilation VariancesPrivateLabels unavailablePrivate saving unavailableQuantum symmetries and continuous generatorsGraduate Quantum Mechanics I · Mixed reviewProgress not loaded
GQM:010231The SU(3) fundamental Casimir is sum Ta^2=4I/3, and exp(i alpha T8) with alpha=4pi/sqrt(3) equals e^{2pi i/3} IGraduate Quantum Mechanics 1 Su3 Quadratic CasimirGraduate Quantum Mechanics 1 Su3 Cartan WeightsPrivateLabels unavailablePrivate saving unavailableQuantum symmetries and continuous generatorsGraduate Quantum Mechanics I · Mixed reviewProgress not loaded
GQM:010322The j=1 Wigner d-matrix at pi/2 maps |1> to amplitudes (1/2, 1/sqrt(2), 1/2) with <Jx>=hbar, <Jy>=0, and Var Jz=hbar^2/2Graduate Quantum Mechanics 1 Wigner D Spin OneGraduate Quantum Mechanics 1 Rotated Angular MomentsPrivateLabels unavailablePrivate saving unavailableAngular momentum, spin, and rotation representationsGraduate Quantum Mechanics I · Mixed reviewProgress not loaded
GQM:010323The j=2 quadrupole HQ=lambda(3Jz^2/hbar^2-6) has energies lambda(6,-3,-6,-3,6); psi=(|2>+2|0>+|-2>)/sqrt(6) survives with amplitude i/3 and probability 1/9 at t=pi hbar/(12 lambda)Graduate Quantum Mechanics 1 Quadrupole Operator SpectrumGraduate Quantum Mechanics 1 Quadrupole Survival AmplitudePrivateLabels unavailablePrivate saving unavailableAngular momentum, spin, and rotation representationsGraduate Quantum Mechanics I · Mixed reviewProgress not loaded
GQM:010327The spin-2 character of Rz(2pi/3) equals sin(5theta/2)/sin(theta/2)=-1, and the C3 average is the rank-1 projector onto |m=0>Graduate Quantum Mechanics 1 Su2 Character FormulaGraduate Quantum Mechanics 1 Cyclic Projector Weight SpacePrivateLabels unavailablePrivate saving unavailableAngular momentum, spin, and rotation representationsGraduate Quantum Mechanics I · Mixed reviewProgress not loaded
GQM:010424The l=0 spherical well with V0=9pi^2/8 (units hbar^2/(2m R^2)) binds at E=-9pi^2/16 with x=y=3pi/4, A^2 R=6pi/(3pi+4), and Pout=2/(3pi+4)Graduate Quantum Mechanics 1 Spherical Well MatchingGraduate Quantum Mechanics 1 Finite Well Exterior ProbabilityPrivateLabels unavailablePrivate saving unavailableBound states and central potentialsGraduate Quantum Mechanics I · Mixed reviewProgress not loaded
GQM:010425The n=2 hard-wall s-wave has E=4 E1, <r>=R/2, <r^2>=R^2(1/3-1/(8 pi^2)), Var=R^2(1/12-1/(8 pi^2)), and P(r<R/2)=1/2View access optionsGraduate Quantum Mechanics 1 Hard Wall Radial MomentsGraduate Quantum Mechanics 1 Hard Wall Half Radius ProbabilityPrivateLabels unavailablePrivate saving unavailableBound states and central potentialsGraduate Quantum Mechanics I · Mixed reviewProgress not loaded
GQM:010526The rank-one Lippmann-Schwinger T-matrix at lambda=1/3, g=1/2 is (2/5)P=(1/5)[[1,1,0],[1,1,0],[0,0,0]], with Born-through-3 coefficient 43/108 and remainder 1/540View access optionsGraduate Quantum Mechanics 1 Rank One T MatrixGraduate Quantum Mechanics 1 Born Series TruncationPrivateLabels unavailablePrivate saving unavailableScattering theory and partial wavesGraduate Quantum Mechanics I · Mixed reviewProgress not loaded
GQM:010628For A=sigma_x, B=sigma_z and t=pi/4 the Trotter overlap is cos q+sqrt(2) sin q=1.7111779716570126 with q=pi/(2 sqrt 2)View access optionsGraduate Quantum Mechanics 1 Bch Pauli TrotterGraduate Quantum Mechanics 1 Unitary Distance FidelityPrivateLabels unavailablePrivate saving unavailableVariational, perturbative, adiabatic, and semiclassical methodsGraduate Quantum Mechanics I · Mixed reviewProgress not loaded
GQM:010629Magnus terms Omega1=-i(sigma_x+sigma_z) and Omega2=-i sigma_y/3 give radii sqrt(2), sqrt(19)/3 and P2=0.5190422230065088View access optionsGraduate Quantum Mechanics 1 Magnus Two TermGraduate Quantum Mechanics 1 Driven Two Level TransitionPrivateLabels unavailablePrivate saving unavailableVariational, perturbative, adiabatic, and semiclassical methodsGraduate Quantum Mechanics I · Mixed reviewProgress not loaded
GQM:010630The Gaussian variational energy of p^2/2+|x| is 3 alpha*/4=0.8128891052089335 at alpha*=(4/pi)^{1/3}View access optionsGraduate Quantum Mechanics 1 Gaussian Variational EnergyGraduate Quantum Mechanics 1 Virial Absolute PotentialPrivateLabels unavailablePrivate saving unavailableVariational, perturbative, adiabatic, and semiclassical methodsGraduate Quantum Mechanics I · Mixed reviewProgress not loaded