Perturbation Theory in Quantum Physics I - Lecture Notes | PHYS 486, Study notes of Quantum Physics

Download Perturbation Theory in Quantum Physics I - Lecture Notes | PHYS 486 and more Study notes Quantum Physics in PDF only on Docsity! Page 1 Physics 486 Lecture 17 Physics 486, Spring ‘07 Lecture 17 Approximation Techniques: Intro. to Perturbation Theory Approximation Methods In many circumstances it may be difficult or inconvenient to exactly solve Schrodinger’s equation – i.e., find the exact eigenstates and eigenvalues - for a quantum particle in a specific potential. To broaden the applicability of quantum mechanics to a variety of important applications, it is important, therefore, to discuss some techniques for approximating the eigenvalues and eigenstates of a Hamiltonian. In the next 2 lectures (Lectures 17 and 18), we will introduce two such techniques: (1) Time-independent (and non-degenerate) perturbation theory (Lecture 17) (2) Variational methods (Lecture 18). We will explore simple examples of these methods here in Phys. 486, but we will investigate more sophisticated applications next semester in Phys. 487. Time-independent Perturbation Theory 2 2 ( ) 2 ( )2 o dH V x m dx ≡ − + For example, one of the key Hamiltonians we have solved exactly in this course is that for the infinite square well potential: V(x)V=∞ V=∞ 0 xL ( ) 2( ) sinon nx x L L π Ψ =     2 2 2 ( ) 22 o n nE mL π =and Suppose we have solved the time-independent SEQ for a particular potential V(x), i.e., by finding the associated eigenstates ψno and eigenvalues, Eno: ( ) ( ) ( ) ( )o o o o n n nH Eψ ψ= ( ) ( )o o n m nmψ ψ δ=where and where Time-independent Perturbation Theory Now, suppose we “perturb” this potential slightly by adding a small addition to the potential, as shown above. The NEW Hamiltonian, H, which is essentially the sum of a solved or “unperturbed” Hamiltonian H(o) PLUS a “perturbing” potential λH’, where we take λ to be a dimensionless and small parameter, can be written as: V(x) V=∞ V=∞ 0 xL ‘perturbation’: λH’ 2 2 ( ) 2' ( ) '2 o dH H H V x H m dx λ λ= + = − + + Now, H has as yet unknown eigenstates ψn(x) and eigenvalues En: n n nH Eψ ψ= (*) Page 2 Physics 486 Lecture 17 Time-independent Perturbation Theory and λ is taken to be a small dimensionless number. We can then write ψn and En as a power series in λ: Where ψn(1) and En(1) are the first-order corrections to the unperturbed eigenstate and eigenvalue, respectively, and ψn(2) and En(2) are the second-order corrections, etc… Plugging these into the time- independent SEQ (*) gives: ( ) ( ) ( ) ( ) (1) 2 (2) ( ) (1) 2 (2) ( ) (1) 2 (2) 'o on n n o o n n n n n n H H E E E λ ψ λψ λ ψ λ λ ψ λψ λ ψ  + + + +   = + + + + + +  Collecting terms with like powers of λ: ( ) (1) 2 (2)o n n n nψ ψ λψ λ ψ= + + + ( ) (1) 2 (2)o n n n nE E E Eλ λ= + + + ( ) ( ) ( ) ( ) ( ) ( ) ( ) (1) ( ) 2 ( ) (2) (1) ( ) ( ) ( ) (1) (1) ( ) 2 ( ) (2) (1) (1) (2) ( ) ' 'o o o o on n n n n o o o o o o n n n n n n n n n n n n H H H H H E E E E E E ψ λ ψ ψ λ ψ ψ ψ λ ψ ψ λ ψ ψ ψ + + + + + = + + + + + + ( ) 'oH H Hλ= +where is the new Hamiltonian To lowest order in λ (i.e., to zeroth order, λ(o)), we just get the unperturbed SEQ: However, to first order in λ (λ(1)), we get: To second-order in λ (λ(2)), we get: ( ) (2) (1) ( ) (2) (1) (1) (2) ( )'o o on n n n n n n nH H E E Eψ ψ ψ ψ ψ+ = + + ( ) ( ) ( ) ( )o o o o n n nH Eψ ψ= ( ) (1) ( ) (1) (1) ( )'o o o on n n n n nH H E Eψ ψ ψ ψ+ = + etc., etc., etc.,… Time-independent Perturbation Theory So, now we can look at successively higher-order contributions to the NEW time-independent SEQ: First-order correction: Energy Let’s look at the first-order correction to the unperturbed energy eigenvalue. Take the inner product of But H(o) is hermitian, so we can write: ⇒ ( ) ( ) (1) ( ) ( ) ( ) ( ) (1) (1) ( ) ( )'o o o o o o o on n n n n n n n n nH H E Eψ ψ ψ ψ ψ ψ ψ ψ+ = + ( ) ( ) (1) ( ) ( ) (1) ( ) ( ) (1) ( ) ( ) (1)o o o o o o o o n n n n n n n n n nH H E Eψ ψ ψ ψ ψ ψ ψ ψ= = = (1) ( ) ( )'o on n nE Hψ ψ= ( ) ( ) 1o on nψ ψ = ( ) ( ) ( )(1) ( )* ( )' 'o on n nE x H x x dx Hψ ψ ∞ −∞ = =∫ ( ) (1) ( ) ( ) (1) (1) ( )'o o o on n n n n nH H E Eψ ψ ψ ψ+ = + with <ψn(o)| (i.e., by multiplying the equation by (ψn(o))* and integrating): ( ) ( ) (1) ( ) ( ) ( ) ( ) (1) (1) ( ) ( )'o o o o o o o on n n n n n n n n n nE H E Eψ ψ ψ ψ ψ ψ ψ ψ+ = + (*) Using the fact that: we get OR: First-order correction: Energy ( ) ( ) ( )(1) ( )* ( )' 'o on n nE x H x x dx Hψ ψ ∞ −∞ = =∫ So, the first-order correction to the energy is: In other words, the first-order correction to the energy is just the expectation value of the perturbation H’ with respect to the unperturbed eigenstates! In matrix language, the first-order energy corrections are the diagonal elements of the matrix H’ computed with respect to a basis set of the unperturbed eigenstates. Page 5 Physics 486 Lecture 17 Supplement: Another Approach to PT Again, imagine that we slightly “perturb” the potential associated with a Hamiltonian, H, whose eigenstates Ψn and eigenvalues En are already known. This NEW Hamiltonian, H’, which is essentially the sum of a solved Hamiltonian H PLUS a “perturbing” potential V’, can be written as: V(x) V=∞ V=∞ 0 xL ‘perturbation’: V’ )(')( 2 '' 2 22 xVxV dx d m VHH ++−=+= Now, H’ has as yet unknown eigenstates ϕn(x) and eigenvalues E’n: nnnn EH ϕϕ '' = Because solving the new SEQ associated with H’ may be difficult, we would like to develop a method of approximating both ϕn(x) and E’n. We start by noting that since our known original eigenstates Ψn form a complete set of functions, we can expand the unknown eigenstates ϕn(x) in a linear combination of Ψm: V(x) V=∞ V=∞ 0 xL ‘perturbation’: V’ Using this expansion, we can rewrite: nnnn EH ϕϕ '' = ( ) ( )xax m m nmn ψϕ ∑ ∞ = = 0 ( ) ( )xExxVxV dx d m nnn ϕϕ '2 22 )(')( 2 =       ++− in the form: ( ) ( )xaExxVxV dx d m a mnm m nm m nm ψψ ∑∑ ∞ = ∞ = =       ++− 0 ' 2 22 0 )(')( 2 Supplement: Another Approach to PT Expanding this out more explicitly: Note that the first term in brackets is simply the unperturbed (i.e., “known”) Hamiltonian. So, we can make the following substitution in the first term: ( ) ( )xaExxVxV dx d m a mnm m nm m nm ψψ ∑∑ ∞ = ∞ = =       ++− 0 ' 2 22 0 )(')( 2 ( ) ( ) ( )xaExxVaxxV dx d m a mnm m nm m nmm m nm ψψψ ∑∑∑ ∞ = ∞ = ∞ = =+       +− 0 ' 0 2 22 0 )(')( 2 nnnn EH Ψ=Ψ ( ) ( )xExxVdx d m nnn ψψ =       +− )( 2 2 22 ( ) ( ) ( )xaExxVaxEa mnm m nm m nmmm m nm ψψψ ∑∑∑ ∞ = ∞ = ∞ = =+ 0 ' 00 )(' Supplement: Another Approach to PT Now, if we multiply both sides of the equation above by Ψ*k, and then integrate both sides over all space, we get: Or, more simply: ( ) ( ) ( )xaEExxVa mnm m mnm m nm ψψ ∑∑ ∞ = ∞ = −= 0 ' 0 )(' ( ) ( ) ( ) ( ) ( )dxxaEExdxxxVax mnm m mnkm m nmk ψψψψ ∑∫∑∫ ∞ = ∞ ∞− ∞ = ∞ ∞− −= 0 '* 0 * )(' ( ) ( ) ( ) ( ) ( )dxxxaEEdxxxVxa mknm m mnmk m nm ψψψψ ∫∑∫∑ ∞ ∞− ∞ = ∞ ∞− ∞ = −= * 0 '* 0 )(' Because the “known” eigenstates Ψn form an orthonormal set ( ) ( ) kmmk dxxx ,* δψψ =∫ ∞ ∞− And we can write (*) as: (*) ( ) ( ) ( ) mknm m mnmk m nm aEEdxxxVxa , 0 '* 0 )(' δψψ ∑∫∑ ∞ = ∞ ∞− ∞ = −= Supplement: Another Approach to PT Page 6 Physics 486 Lecture 17 Or equivalently: Which can then be written: So far, we have made no approximations. Now, let’s assume that the perturbing potential V’ is small compared with the unperturbed potential V. In this case, we expect that: Which means that in our original expansion: Vkm ( ) ( ) ( ) nkknmk m nm aEEdxxxVxa −=∫∑ ∞ ∞− ∞ = '* 0 )(' ψψ ( ) ( ) ( )dxxxVxaEEa mk m nmknnk ψψ )(' * 0 ' ∫∑ ∞ ∞− ∞ = =− ( ) km m nmknnk VaEEa ∑ ∞ = =− 0 ' ( ) ( )xx nn ψϕ ≈ ( ) ( ) ( )xxax nm m nmn ψψϕ ≈= ∑ ∞ =0 1≈nna 1<<nmaand (n ≠ m) Supplement: Another Approach to PT So, we can write the following “first-order” approximation to the equation: which is: Therefore, ( ) ( ) ( )dxxxVxaEEa mk m nmknnk ψψ )(' * 0 ' ∫∑ ∞ ∞− ∞ = =− and ( ) knnnkm m nmknnk VaVaEEa ≈=− ∑ ∞ =0 ' ( ) knknnk VEEa ≈−' (using ann ~ 1) ( ) ( ) ( ) ( )knnknkkn EEadxxxVxV −≈= ∫ ∞ ∞− ''* ψψ kn kn kn a VEE ≈−' Now, if we let k=n, then ( ) ( ) ( )dxxxVxV a VEEE nnnn nn nn nn ψψ '*' ∫ ∞ ∞− =≈≈−=∆ Thus, the “first-order correction” to the energy eigenstates of the “unperturbed” Hamiltonian, due to the perturbing potential V’, are ( ) ( ) ( ) ''* VdxxxVxE nn =≅∆ ∫ ∞ ∞− ψψ which is again the expectation value of the perturbing potential with respect to the unperturbed eigenstates! Supplement: Another Approach to PT We can also determine the approximate eigenstates of the “perturbed” Hamiltonian, by calculating the “first-order” corrections to the “unperturbed” eigenstates due to the perturbing potential V’. Consider the expansion: So, from our result for the first-order correction to the energy eigenvalues: ( ) ( ) ( )* ' 'nn n nE V x V x x dx Vψ ψ ∞ −∞ ∆ = = =∫where ( ) ( )xax m m nmn ψϕ ∑ ∞ = = 0 ( ) knknnk VEEa ≈−' ( )kn kn nk EE Va − ≈ ' ' n n n nnE E E E V≈ + ∆ = + Recall that we made the approximation ann ≈ 1 in this expansion, which gave the result: Supplement: Another Approach to PT Combining ( )kn kn nk EE Va − ≈ ' ' n n nnE E V≈ + which can be written: and gives: ( ) ( ) ( )            − +− =       − −+ − = −+ ≈ kn nnkn kn kn knnn kn kn knnn kn nk EE VEE V EE EVEEE V EVE Va 1 1 ( ) 1 1 −       − + − ≈ kn nn kn kn nk EE V EE Va Supplement: Another Approach to PT Page 7 Physics 486 Lecture 17 Now, since Vnn << En - Ek, i.e., we have assumed a small perturbing potential V’, we can expand the term to the right in parenthesis in a binomial expansion, and drop terms in the expansion higher than first-order: ( ) ( )      − − − ≈      − + − ≈ − kn nn kn kn kn nn kn kn nk EE V EE V EE V EE Va 11 1 ( )kn kn nk EE Va − ≈ This is the first-order approximation to the coefficients in the expansion: ( ) ( )xax m m nmn ψϕ ∑ ∞ = = 0 ( ) ( ) ( ) ( )∑ ∞ ≠ − +≈ nm m mn nm nn xEE Vxx ψψϕ Supplement: Another Approach to PT