Ch6 - Approximation method

Ch6 - Approximation method - Chapter 6 Approximation...

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Approximation methods for stationary problems Chapter 6 Reference: Bransden and Joachain Chapter 7
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Time independent perturbation theory (non-degenerate systems) 0 ' H H H ε = + (0) 0 nn n HE ψψ = An unperturbed system has a Hamiltonian , and its eigenvectors and eigenvalues are known: 0 H The system is perturbed slightly by adding to the Hamiltonian, so that the new system Hamiltonian is given by ' H Questions: What are eigenvalues and eigenvectors of the new H ? Is there a systematic way to solve the eigenvalue problem When is sufficiently small? What do we mean small? () 3 H a a a a ω + + + + = e.g. ε = dimensionless small number nm EE n m nonlinear
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() 0 kk nn k ψε ψ = = 00 '' H HH H V ε = += + n HE = Our goal is to solve with Let 0 0 ' j j n kj k E εε ∞∞ == = ⎛⎞ ⎜⎟ ⎝⎠ ∑∑ 0 k EE = = First order in terms: (1) (0) (0) (0) 0 ' n n n n HHE E ψψ + (0) 0 ' n n n n n n n n E + ' nnn n n n EH E E + (0) (0) ' n = First order energy correction of the level n n n HV .... n E = ++
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(1) (0) (0) (0) 0 ' nn n n n n HHE E ψψ ψ += + (0) 0 (' ) n n HE E H ⎡⎤ −= ⎣⎦ (0) m m m c = First order correction to the eigenvector : Let ( 0 )( 1 ) ( 1 0 ) 1 ... n n n n εψ ψ =+ + + i n e α ' mn nm H cn m EE = −≠ Require non-degenerate energies () 0 kk k ψε = = We can take , because if it is non-zero, it only affects the overall phase 0 nn c = * 1. . . 1 nn nn cc εε + normalization (to first order) would force , a purely imaginary number nn ci θ = But is also an eigenvector for any overall phase angle
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( 2 ) ( 1 ) ( 0 )( 2 ) ( 1 1 ) ( 2 0 ) 0 ' nn n n n n n n HH E E E ψ ψψψψ += + + 2 ε Second order terms : (2) (0) (1) ' n n n n EH E ψψ =− (0) m m m c = ' mn nm H nmc EE ≠= ' H = ' nm n m m cH = 2 | ' | m H Second order energy correction to the level n 2 2( 2 ) | ' | n nm V E ==
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Summary of first and second order corrections (non-degenerate systems) (0) ' .... mn nn m nm V EE ψψ ψ =+ + 0 ' H HV = + n HE = 2 |' | ' ... n n V V + + (1) n E Δ (2) n E Δ ±²³²´ ±²²²³²²²´ n ±²²²³ ² ² ²´
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x = 0 x = a x = a/2 0 ,0 0/ 2 2 0 xa x VV x a ax ∞>< =< < < < (0) 2 () s i n n nx x aa π ψ = 22 2 2 2 n n E ma = = Example: Perturbation inside a 1D infinite square well Unperturbed states: /2 (1) (0)* 0 0 '( ) ( ) a nn n n n EV x V x d x ψψ Δ= = First order correction of eigen-energies 2 0 0 2 sin a Vn x dx ⎛⎞ = ⎜⎟ ⎝⎠ 0 2 V = n =1,2,….
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Example: Relativistic correction to kinetic energy 24 22 2 32 2 8 pp Tm c p c m c m mc =+ Question: What is the first order correction to the ground state energy of an SHO? 4 (1) (0) (0) ', ' 8 nn n p EV V ψψ Δ= = 4 * 00 0 () 8 p Ex x d x ⎛⎞ ⎜⎟ ⎝⎠ For the ground state, 2 1/4 /2 0 mx m xe ω ψ π = = = 2 15 32 =− = d pi dx =
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Example: Two-level systems 1 2 eg H ge = 21 e < <− (0) 1 1 0 ψ = 2 0 1 = 0 ' 0 g V g = Unperturbed states (zero order states): Perturbed part of the Hamiltonian: First order correction of the energies: (1) (0) (0) '0 jj j EV ψψ Δ == ee > assumed j =1,2
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(0) 2 (2) | ' | mn n nm V E EE ψψ Δ= 2 2 21 1 12 |' | V g E ee = Second order correction to the energies: 2 2 2 | V g E = Comparing with the exact eigenvalues: 2 22 2 11 () 4 1 4 g e e g e e λ ±
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Ch6 - Approximation method - Chapter 6 Approximation...

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