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By Oppenheimer J. R.

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A1;N 6 a2;1 a2;2 a2;3 . . . a2;N 7 7 6 6 a3;1 a3;2 a3;3 . . . a3;N 7 7 6 ð2-21Þ 6 ... ... 7 7 6 4 ... ... 5 aM;1 aM;2 aM;3 . . . aM;N 28 THE MATHEMATICS OF QUANTUM MECHANICS An abbreviated notation for a matrix is ½ai; j Š i ¼ 1; 2 . . M j ¼ 1; 2 . . N ð2-22Þ or simply ½AŠ or A ð2-23Þ We note that a vector is represented by a boldface lowercase letter, whereas a matrix is described by a boldface capital letter in square brackets or by a boldface capital letter alone. The notation (2-23) is allowed only after the matrix has been defined in more detail.

23 24 THE MATHEMATICS OF QUANTUM MECHANICS Sommerfeld continued his mathematics studies with Felix Klein (1849–1925) in Go¨ ttingen. In addition to being an outstanding mathematician, Klein was a firstrate administrator and politician, and at the time he was probably the most influential mathematician in the country. Sommerfeld became Klein’s star student and he tackled one of the more challenging problems in mathematical physics, the motion of the gyroscope. Somerfeld’s elegant solution of the problem, which was published between 1897 and 1910, was considered a major contribution to the field of mathematics.

A2;N xN ¼ 0 a3;1 x1 þ a3;2 x2 þ ða3;3 À lÞx3 þ . . . þ a3;N xN ¼ 0 ... ... ð2-61Þ ... ¼ 0 aN;1 x1 þ aN;2 x2 þ aN;3 x3 þ . . . þ ðaN;N À lÞxN ¼ 0 This is a set of N homogeneous equations with N unknowns, and according to our previous discussion, this set of equations will have a solution if and only if the determinant of the coefficients Á is equal to zero. Let us now proceed to the eigenvalue problem of the matrix ½ai; j Š. The problem is defined as the derivation of its eigenvalues and eigenvector.

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