Problems related to eigenvalue equations
1. Determine which of the following functions are eigenfunctions to the operator d
dx
2
(a): e ikx ; (b): cos(kx) ; (c) : k ; (d): kx ; (e): e − x
Give the corresponding eigenvalue where appropriate
Answer:
In each case form Ωf . If the result is f where
of the operator Ω and
is the eigenvalue
(a):
de ikx
ikx
= ike , yes: eigenvalue =
dx
is a constant, then f is an eigenfunction
ik
(b): d cos(kx) = −k sin kx; no
dx
(c): dk = 0; yes; eigenvalue 0
dx
(d): dkx = k = 1 kx; no [ 1 is not a constant]
dx
x
x
2
de − x
(e):
= −2 xe − x ; no [ − 2 x is not a constant]
dx
2
2. Determine which of th following functions are eigenfunctions of the inversion operator
iˆ (which has the effect of making the replacement x → - x.
Answer :
( a ) : x3 − kx ; ( b ) : coskx ; (c):x 2 + 3x −1.
State the eigenvalue of iˆ when appropriate
Operate on each function with iˆ ; if the function is
regenerated multiplied by a constant, it is an eigenfunction
of iˆ and the constant is the eigenvalue.
( b ) : f = coskx ; (c)
iˆcoskx = cosk(-kx) = coskx = f
Therefore, f is an eigenfunction with eigenvalue, +1
( c ) : f = x 2 + 3x −1
iˆ (x 2 + 3x − 1 ) =x 2 − 3x −1 ≠ (constant)*f
Therefor, f not an eigenfunction to iˆ
2
3. 1. Determine which of the following functions are eigenfunctions to the operator d
dx 2
2
(a): e ikx ; (b): cos(kx) ; (c) : k ; (d): kx ; (e): e − x
Give the corresponding eigenvalue where appropriate
In each case form Ωf . If the result is f where
of the operator Ω and
is the eigenvalue
is a constant, then f is an eigenfunction
Answer:
2
ikx
(a): d (e ) = − k2 e ikx , y e s :eigenvalue = −k 2
dx 2
2
(b): d cos(kx) = − k2 cos kx; yes : eigenvalue= - k 2
dx2
2
(c): d k = 0; yes; eigenvalue 0
dx 2
2
(d): d (kx) = 0 = 0(kx); yes eigenvalue 0
dx 2
2 − x2
d
e
2 2 − x2
(e):
=
(−2
+
4
x )e
; no
dx 2
d2
;
dx 2
d
d2
, but not of
.
(b,d) are eigenfunctions of
2
dx
dx
Hence (a,b,c,d) are eigenfunctions of