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marina-yarberry | 2014-12-14 | General

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Problems related to eigenvalue equations 1. Determine which of the following functions are eigenfunctions to the operator dx (a): ikx ; (b): cos( kx ; (c) : ; (d): kx ; (e): Give the corresponding eigenvalue where appropriate Answer: In each case form . If the result is where is a constant, then is an eigenfunction of the operator and is the eigenvalue (a): de ikx dx ike ikx yes: eigenvalue ik (b): cos( kx dx sin kx ; no (c): dk dx yes; eigenvalue 0 (d): dkx dx kx no [ is not a constant] (e): de dx xe ; no [ is not a constant]

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2. Determine which of th following functions are eigenfunctions of the inversion operator i (which has the effect of making the replacement x -x Answer : (a): x kx ; (b): coskx ; (c): x State the eigenvalue of i when appropriate Operate on each function with i ; if the function is regenerated multiplied by a constant, it is an eigenfunction of i and the constant is the eigenvalue. (c): f 1 i 1 )= 1 constant)* f Therefor, f not an eigenfunction to i (b): f coskx ; (c) i coskx cosk(-kx) coskx Therefore, f is an eigenfunction with eigenvalue, +1

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3. 1. Determine which of the following functions are eigenfunctions to the operator dx (a): ikx ; (b): cos( kx ; (c) : ; (d): kx ; (e): Give the corresponding eigenvalue where appropriate In each case form . If the result is where is a constant, then is an eigenfunction of the operator and is the eigenvalue Answer: (a): ikx dx ikx , yes: eigenvalue (b): cos( kx dx cos kx ; yes : eigenvalue (c): dx 0; yes; eigenvalue 0 (d): kx dx 0( kx ); yes eigenvalue 0 (e): dx ; no Hence (a,b,c,d) are eigenfunctions of dx are eigenfunctions of dx but not of dx

Determine which of the following functions are eigenfunctions to the operator dx a ikx b cos kx c d kx e Give the corresponding eigenvalue where appropriate Answer In each case form If the result is where is a constant then is a ID: 24012

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Page 1

Problems related to eigenvalue equations 1. Determine which of the following functions are eigenfunctions to the operator dx (a): ikx ; (b): cos( kx ; (c) : ; (d): kx ; (e): Give the corresponding eigenvalue where appropriate Answer: In each case form . If the result is where is a constant, then is an eigenfunction of the operator and is the eigenvalue (a): de ikx dx ike ikx yes: eigenvalue ik (b): cos( kx dx sin kx ; no (c): dk dx yes; eigenvalue 0 (d): dkx dx kx no [ is not a constant] (e): de dx xe ; no [ is not a constant]

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2. Determine which of th following functions are eigenfunctions of the inversion operator i (which has the effect of making the replacement x -x Answer : (a): x kx ; (b): coskx ; (c): x State the eigenvalue of i when appropriate Operate on each function with i ; if the function is regenerated multiplied by a constant, it is an eigenfunction of i and the constant is the eigenvalue. (c): f 1 i 1 )= 1 constant)* f Therefor, f not an eigenfunction to i (b): f coskx ; (c) i coskx cosk(-kx) coskx Therefore, f is an eigenfunction with eigenvalue, +1

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3. 1. Determine which of the following functions are eigenfunctions to the operator dx (a): ikx ; (b): cos( kx ; (c) : ; (d): kx ; (e): Give the corresponding eigenvalue where appropriate In each case form . If the result is where is a constant, then is an eigenfunction of the operator and is the eigenvalue Answer: (a): ikx dx ikx , yes: eigenvalue (b): cos( kx dx cos kx ; yes : eigenvalue (c): dx 0; yes; eigenvalue 0 (d): kx dx 0( kx ); yes eigenvalue 0 (e): dx ; no Hence (a,b,c,d) are eigenfunctions of dx are eigenfunctions of dx but not of dx

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