2. We consider the harmonic oscillator in one dimension as discussed in section 1.3.2. The time-independent Schrödinger equation is +mw²²(x) = Ev(x). 2m dar2 You are given 1 a√ 1 e-2²/(2a²) =√√20√* (2) h with a = m.w It is useful to use the results from question 1 below. (a) Show that o(r) is correctly normalised. (b) Show that u₁(r) is correctly normalised. (c) Show that o(a) solves the Schrödinger equation, with E = hw. (1
2. We consider the harmonic oscillator in one dimension as discussed in section 1.3.2. The time-independent Schrödinger equation is +mw²²(x) = Ev(x). 2m dar2 You are given 1 a√ 1 e-2²/(2a²) =√√20√* (2) h with a = m.w It is useful to use the results from question 1 below. (a) Show that o(r) is correctly normalised. (b) Show that u₁(r) is correctly normalised. (c) Show that o(a) solves the Schrödinger equation, with E = hw. (1
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Transcribed Image Text:2. We consider the harmonic oscillator in one dimension as discussed in section 1.3.2. The time-independent Schrödinger
equation is
h² d² √(x)
2m dr2
+ ½ mw²¹² y(x) = Ev(x).
You are given
1
-x²/(2a²),
Vo(x) =
a√T
1
e-r²/(24²)
(2 a√F (Za)
h
with a =
m.w
It is useful to use the results from question 1 below.
(a) Show that
(r) is correctly normalised.
(b) Show that
(r) is correctly normalised.
(c) Show that
o(a) solves the Schrödinger equation, with E= Shw.
(d) Show that ₁ (r) solves the Schrödinger equation, with E-hw.
=
(e) Show that fx dx (x) ₁(x) = 0, i.e. the different eigenfunctions are orthogonal.
(f) Plot o(r) and ₁(r).
e
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