3. In momentum space the Schrödinger equation reads, ap(p.t) p² 2μ Ət ih- = -P(p. t) + V (-1/20p) e(P.1). Op (a) Show that the time dependence of the wave function ,(p, t) of a bound state with the energy E, can be factorized, thereby transforming (Eq. 3a) into a time independent equation. (5) (b) For the oscillator potential, given in coordinate space by V(r) = μw²r²/2, show that the time-independent equation found in (a) is of the form. w²y²₂(y) - n (p): (Eq. 3a) E₂9₂ (y) = (Eq. 3b) (8) where p = pwy and o, (y) = (y). (c) Since (Eq. 3b) has the same form as the time-independent Schrödinger equation for the oscillator in coordinate space, we can conclude that ,, (y) may differ from (r) only by normalization. Hence, show that - ħ² d² 2μ dy2 Pn (y), elan View (²). where (r) is the oscillator eigenfunction in coordinate space. (12) [25] TURN OVER]
3. In momentum space the Schrödinger equation reads, ap(p.t) p² 2μ Ət ih- = -P(p. t) + V (-1/20p) e(P.1). Op (a) Show that the time dependence of the wave function ,(p, t) of a bound state with the energy E, can be factorized, thereby transforming (Eq. 3a) into a time independent equation. (5) (b) For the oscillator potential, given in coordinate space by V(r) = μw²r²/2, show that the time-independent equation found in (a) is of the form. w²y²₂(y) - n (p): (Eq. 3a) E₂9₂ (y) = (Eq. 3b) (8) where p = pwy and o, (y) = (y). (c) Since (Eq. 3b) has the same form as the time-independent Schrödinger equation for the oscillator in coordinate space, we can conclude that ,, (y) may differ from (r) only by normalization. Hence, show that - ħ² d² 2μ dy2 Pn (y), elan View (²). where (r) is the oscillator eigenfunction in coordinate space. (12) [25] TURN OVER]
Related questions
Question
![3. In momentum space the Schrödinger equation reads,
ap(p.t) p²
2μ
Ət
ih-
=
-P(p. t) + V
(-1/20p) e(P.1).
Op
(a) Show that the time dependence of the wave function ,(p, t) of a bound state
with the energy E, can be factorized, thereby transforming (Eq. 3a) into a
time independent equation.
(5)
(b) For the oscillator potential, given in coordinate space by V(r) = μw²r²/2, show
that the time-independent equation found in (a) is of the form.
w²y²₂(y) -
n (p):
(Eq. 3a)
E₂9₂ (y) =
(Eq. 3b)
(8)
where p = pwy and o, (y) = (y).
(c) Since (Eq. 3b) has the same form as the time-independent Schrödinger equation
for the oscillator in coordinate space, we can conclude that ,, (y) may differ from
(r) only by normalization. Hence, show that
-
ħ² d²
2μ dy2 Pn (y),
elan
View
(²).
where (r) is the oscillator eigenfunction in coordinate space.
(12)
[25]
TURN OVER]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc904c1d-eb70-486e-99cb-d3443db68783%2F43247818-8b6d-4b34-9c04-ab870e6007f3%2Ftk37f9l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. In momentum space the Schrödinger equation reads,
ap(p.t) p²
2μ
Ət
ih-
=
-P(p. t) + V
(-1/20p) e(P.1).
Op
(a) Show that the time dependence of the wave function ,(p, t) of a bound state
with the energy E, can be factorized, thereby transforming (Eq. 3a) into a
time independent equation.
(5)
(b) For the oscillator potential, given in coordinate space by V(r) = μw²r²/2, show
that the time-independent equation found in (a) is of the form.
w²y²₂(y) -
n (p):
(Eq. 3a)
E₂9₂ (y) =
(Eq. 3b)
(8)
where p = pwy and o, (y) = (y).
(c) Since (Eq. 3b) has the same form as the time-independent Schrödinger equation
for the oscillator in coordinate space, we can conclude that ,, (y) may differ from
(r) only by normalization. Hence, show that
-
ħ² d²
2μ dy2 Pn (y),
elan
View
(²).
where (r) is the oscillator eigenfunction in coordinate space.
(12)
[25]
TURN OVER]
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