Fund Quantum Mechanics Lect & HW Solutions 30

Fund Quantum Mechanics Lect & HW Solutions 30 - d x...

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12 CHAPTER 2. MATHEMATICAL PREREQUISITES 2.5 Eigenvalue Problems 2.5.1 Solution eigvals-a Question: Show that e i kx , above, is also an eigenfunction of d 2 / d x 2 , but with eigenvalue k 2 . In fact, it is easy to see that the square of any operator has the same eigenfunctions, but with the square eigenvalues. Answer: DiFerentiate the exponential twice, [1, p. 60]: d d x e i kx = i ke i kx d d x p d d x e i kx P = d d x ± ike i kx ² = (i k ) 2 e i kx So d 2 / d x 2 turns e i kx into (i k ) 2 e i kx ; the eigenvalue is therefore (i k ) 2 which equals k 2 . 2.5.2 Solution eigvals-b Question: Show that any function of the form sin( kx ) and any function of the form cos( kx ), where k is a constant called the wave number, is an eigenfunction of the operator d 2 / d x 2 , though they are not eigenfunctions of d
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Unformatted text preview: / d x Answer: The ±rst derivatives are, [1, p. 60]: d d x sin( kx ) = k cos( kx ) d d x cos( kx ) = − k sin( kx ) so they are not eigenfunctions of d / d x . But a second diFerentiation gives: d d x p d d x sin( kx ) P = d d x ( k cos( kx )) = − k 2 sin( kx ) d d x p d d x cos( kx ) P = d d x ( − k sin( kx )) = − k 2 cos( kx ) 2.5.3 Solution eigvals-c Question: Show that sin( kx ) and cos( kx ), with k a constant, are eigenfunctions of the inversion operator Inv, which turns any function f ( x ) into f ( − x ), and ±nd the eigenvalues....
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