Methods of Approximation PDF

METHODS OF APPROXIMATION Time-independent perturbation theory (non-degenerate) Time-independent degenerate perturbation theory Variational method Time-dependent perturbation theory  DEGENERATE PERTURBATION THEORY Why is degenerate perturbation theory needed? 2-fold...

METHODS OF APPROXIMATION Time-independent perturbation theory (non-degenerate) Time-independent degenerate perturbation theory Variational method Time-dependent perturbation theory  DEGENERATE PERTURBATION THEORY Why is degenerate perturbation theory needed? 2-fold degeneracy – Zero-order eigenfunctions – First-order energies Generalization to ζ –fold degeneracy Example: the Stark effect in the hydrogen atom – Ground state (non-degenerate) – First excited state (degenerate)  SUMMARY - PERTURBATION THEORY The Hamiltonian of the A small time-independent system we are interested in perturbing potential ˆ ˆ ˆ ˆ ˆ H  H 0  H p  H 0   H1 λ is a measurement of the smallness of the perturbation. It is a device to A Hamiltonian we can keep track of how large terms in the solve exactly equations are.  NON-DEGENERATE PERTURBATION THEORY SUMMARY - PERTURBED ENERGIES & STATES En  En(0)   En(1)   2 En(2) ...  En0   En(1)   En(2) ... 2 k  Hˆ 1 n  En0  n  Hˆ 1 n   ... k n E E 0 n 0 k  n   n( 0)   n(1)   2 n(2) ...  n   n(1)   n(2) ... k  Hˆ 1 n  n   k k n E E 0 n 0 k    Hˆ    ˆ H    ˆ H    ˆ H    n    m  k 0 1 n0 m0 1 k  n 1 n m 1  ... mn  k  n  En  Ek En  Em0   En0  Em0 2     If there is degeneracy, i.e. En0  Ek0 for n  k , then 2 k  Hˆ 1 n k  Hˆ 1 n  n(1)   k and  En(2)   k n En0  Ek0 k n En0  Ek0 give meaningless results. DEGENERATE PERTURBATION THEORY is needed! EXAMPLE: 2-FOLD DEGENERACY The unperturbed n th state is 2-fold degenerate Hˆ 0n  En0n Hˆ 0n  En0n n n   ,  any linear combination of n and n is also an eigenstate of Hˆ 0 with the same eigenvalue  n  c n  c n c , c     Hˆ 0  n  Hˆ 0 c n  c n  c Hˆ 0n  c Hˆ 0n  c En0n  c En0n  En0 c   n   c n  En0  n  From previous lecture THE PERTURBED SYSTEM Hˆ 0 n( 0)  En0 n(0) ˆ (0) ˆ H1 n  H 0 n  En  n  En n (1) (1) (0) 0 (1) ˆ ˆ H1 n  H 0 n  En  n  En  n  En n (1) (2) (2) ( 0) (1) (1) 0 (2)  2-fold degeneracy ZERO-ORDER EIGENFUNCTIONS In non-degenerate perturbation theory  n(0)  n In degenerate perturbation theory  n(0)  c n  c n 2 2  (0) n  (0 ) n  c  c  1 Hˆ 0 n(0)  En0 n(0) Hˆ  (0)  Hˆ  (1)  E (1) (0 )  E 0 (1) 1 n 0 n n n n n  2-fold degeneracy FIRST-ORDER ENERGY CORRECTIONS Hˆ 1 n(0)  Hˆ 0 n(1)  En(1) n(0)  En0 n(1)    Hˆ 0  En0  n(1)  En(1)  Hˆ 1  n(0)   Hˆ 0  En 0  (1) n  E(1) n  Hˆ   c  1  n  c n  bracket with n    n Hˆ 0  En0  n(1)  n En(1)  Hˆ 1 c n  c n       c n En(1)  Hˆ 1 n  c n En(1)  Hˆ 1 n  2-fold degeneracy FIRST-ORDER ENERGY CORRECTIONS      n Hˆ 0  En0  n(1)  c n En(1)  Hˆ 1 n  c n En(1)  Hˆ 1 n  n Hˆ 0  n(1)  n En0  n(1)  c En(1) n n  c n Hˆ 1 n  c En(1) n n  c n Hˆ 1 n       1 0 n Hˆ 0  n(1)  n En0  n(1)  c En(1)  c n Hˆ 1 n  c n Hˆ 1 n Hˆ 0n  n(1)  En0 n  n(1)  c En(1)  c n Hˆ 1 n  c n Hˆ 1 n En0 n  n(1)  En0 n  n(1)  c En(1)  c n Hˆ 1 n  c n Hˆ 1 n    0  0  c En(1)  n Hˆ 1 n c  n Hˆ 1 n   c n Hˆ 1 n  En(1)  c n Hˆ 1 n  0  2-fold degeneracy FIRST-ORDER ENERGY CORRECTIONS  Hˆ  E    E  Hˆ   c   c   0 0 n (1) n (1) n 1  n  n bracket with n     n Hˆ 0  En0  n(1)  c n En(1)  Hˆ 1 n  c n En(1)  Hˆ 1 n   En0 n  n(1)  En0 n  n(1)  c n Hˆ 1 n  c En(1)  c n Hˆ 1 n    0 c n Hˆ 1 n  c  n  Hˆ 1 n  En(1)  0 to be considered together with the equation from the previous slide i.e.   c n Hˆ 1 n  En(1)  c n Hˆ 1 n  0  2-fold degeneracy FIRST-ORDER ENERGY CORRECTIONS  n 1 n n  c  Hˆ   E (1)  c  Hˆ   0   n 1 n    c n Hˆ 1 n  c n Hˆ 1 n  En(1)  0  Matrix elements of the perturbation n Hˆ 1 n  H1   c H1  En(1)  c H1  0   c H   1  c H1  E (1) n 0  2-fold degeneracy FIRST-ORDER ENERGY CORRECTIONS   c H1  En(1)  c H1  0   system of linear equations  c H1  c H1  En  0 (1)  non-vanishing solutions for c and c if and only if H1  En(1) H1 0 H1 H1  E (1) n  H  E  H  E   H 1 (1) n 1 (1) n 1 H1    0 E  H  H  E  H H 2 (1) n 1  1 (1) n 1 1  H1 H1  0  2-fold degeneracy FIRST-ORDER ENERGY CORRECTIONS H1  En(1) H1 0 H1 H1  En (1) This is equivalent to the calculation of the eigenvalues (En(1) ) of the matrix  H1 H1  H H1   1  2-fold degeneracy FIRST-ORDER ENERGY CORRECTIONS   c H1  En(1)  c H1  0    c H1  c H1  En  0 (1)    H1 H1  c  (1)    c     En   H  1 H1  c  c    2-fold degeneracy FIRST-ORDER ENERGY CORRECTIONS   H  H1  En(1)  H1 H1  H1 H1  0 2 En(1) 1 2nd order equation with, in general, two different real solutions for En(1) : En(1)1 and En(1)2. En(1)1 and En(12 ) are in general different: the eigenvalues of Hˆ are not degenerate. The 2-fold degenerate level En0 of the unperturbed system is split into 2 different levels: En1  En0   En(1)1 En2  En0   En(1)2 The perturbation has lifted the degeneracy (in general the lifting may be either total or partial for  -fold degeneracy)  2-fold degeneracy ZERO-ORDER EIGENFUNCTIONS (1) (1)   c H1  En(1)  c H1  0  By substituting En1 and En2 into  c H   1  c  H 1   En (1) 0   2 sets of c and c are found, hence 2 zero-order eigenfunctions which, in general, correspond to different first-order energy corrections.  n(0)  c n  c n  En  En0   En(1) 1 1 1 1 1  n(0)  c n  c n  En  En0   En(1) 2 2 2 2 2 Normalization: 2 2 2 2  (0) n1  (0) n1  c1  c1 1  (0 ) n2  (0) n2  c 2  c2 1  2-fold degeneracy Working with Hp and δE(1) rather than H1 and E(1)   c H1  En(1)  c H1  0     c  H1   En(1)  c  H1  0       c H1  c H1  En  0 (1)  c  H1  c  H1   En  0 (1)   En(1)   En(1)  Hˆ 1  Hˆ p  H1  H p    c H p   En(1)  c H p  0  H p H1  c  (1)    c       En     c H p  c H p   En  0 (1)  H p H 1     c   c  En  En0   En(1)