Section III, Quantum Physics II
Remainder on Pauli matrixes; Variational Methods
1) Let σk, k = 1 , 2 , 3 stand for the Pauli matrixes. Show that [σa, σb] = 2iabcσc,
where abc means the Levi-Civity symbol. Let {σa, σb} stand for the anti- commutator, i.e. {σa, σb} := σaσb+ σbσa. Show
{σa, σb} = 2δabI ,
where δab denotes the Kronecker delta and I unit matrix.
Estimation of ground state energy. The ground state energy E0 of the system governed by the Hamiltonian H satisfies
E0 = min
kψk=1hψ|Hψi, .
Relying on this fact and choosing a test function solve the following problems.
2) Estimate the ground state energy of the hydrogen atom; as a test function choose
Aexp(−λ(r/aB)) . (0.1)
3) Consider an electron moving in three dimensional system under an influ- ence of the screening Coulomb potential
−V0exp(−r/rD)/r ;
where rD denotes a constant. Using the variational methods find an esti- mation on ground state energy. As a test function you can an exponential function analogous to (0.1).
4) Consider a particle moving in one dimensional system under an influence of the oscillator potential, i.e. the Hamiltonian takes the form
H = −~2 2m
d2 dx2 +1
2mω2x2, ω = const . 1
Choosing properly a test function show that E0 ≤ 1
2~ω .
Actually, the right hand side of the above inequality gives the exact ground state energy.
ps. If you want to choose properly a test function analyse the behaviour of the potential.
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