Chapter 7. The Eigenvalue Problem

More documents

Recommendations

Info

7. The Eigenvalue Problem, December 17, 2009 6 Normally in quantum mechanics we choose the basis set {|a 1 ,...,|a n } and know the operator Â. This means that we can calculate A mn = a m | Â | a n. We do not know |ψ or λ, and our mission is to find them from Â|ψ = λ|ψ. We have converted this operator equation, by using the orthonormal basis set {|a 1 ,...,|a N }, into the matrix equation Eq. 23 (or Eq. 25 or Eq. 26). In Eq. 23, we know A mn but we do not know ψ = {ψ 1 , ψ 2 ,...,ψ N } or λ. WemustcalculatethemfromEq.23.Onceweknowψ 1 , ψ 2 ,...,ψ N ,wecan calculate |ψ from |ψ = N |a n a n | ψ ≡ n=1 N ψ n |a n (27) § 6 Use a computer. Calculating eigenvalues and eigenvectors of a matrix “by hand” is extremely tedious, especially since the dimension N is often very large. Most computer languages (including Fortran, C, C++, Mathematica, Maple, Mathcad, Basic) have libraries of functions that given a matrix will return its eigenvalues and eigenvectors. I will not discuss the numerical algorithms used for finding eigenvalues. Quantum mechanics could leave that to experts without suffering much harm. The Mathematica file Linear algebra for quantum mechanics.nb shows how to use Mathematica to perform calculations with vectors and matrices. Nevertheless, it is important to understand some of the theory related to the eigenvalue problem, since much of quantum mechanics relies on it. I presentsomeofitinwhatfollows. § 7 The eigenvalue problem and systems of linear equations. The eigenvalue problem Eq. 23 can be written as n=1 or ⎛ ⎜ ⎝ N (A mn − λδ mn )ψ n =0, m =1, 2,...,N (28) n=1 A 11 − λ A 12 A 13 ··· A 1N A 21 A 22 − λ A 23 ··· A 2N A 31 A 32 A 33 − λ ··· A 3N . . . . . . . A N1 A N2 A N3 ··· A NN − λ ⎞ ⎛ ⎟ ⎜ ⎠ ⎝ ψ 1 ψ 2 ψ 3 . ψ N ⎞ ⎛ = ⎟ ⎜ ⎠ ⎝ 0 0 0. 0 ⎞ ⎟ ⎠ d = 0 (29)
7. The Eigenvalue Problem, December 17, 2009 7 The column in the right-hand side is the zero vector. We can regard δ mn asbeingtheelementsofamatrixI, which is called the unit matrix (or the identity matrix): ⎛ ⎞ 1 0 0 ······ 0 0 1 0 ······ 0 I = 0 0 1 ······ 0 (30) ⎜ . ⎝ . . . . . . ⎟ ⎠ 0 0 0 ······ 1 This has the property (Iv) m = N I mn v n = n=1 N δ mn v n = v m n=1 The m-th element of the vector Iv is v m ;thismeansthat The definition of I allows us to write Eq. 29 as A system of equations of the form Iv = v (31) (A − λI)ψ = 0 (32) Mv = 0, (33) where M is an N × N matrix and v is an N-dimensional vector, is called a homogeneous system of N linear equations. The name ‘homogeneous’ is given because the right-hand side is zero; this distinguishes it from Mv = u (34) which is an inhomogeneous system of N linear equations. Example. Let A = ⎛ ⎜ ⎝ 3 2 4 2 1.2 3.1 4 3.1 4 ⎞ ⎟ ⎠ (35)
Page 1 and 2: 7. The Eigenvalue Problem, December
Page 5: 7. The Eigenvalue Problem, December

Chapter 7. The Eigenvalue Problem

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?