Problem 2.11 Show that the lowering operator cannot generate a state of infinite norm (i.e., f la.-²dx < equation). What does this tell you in the case y = vo? Hint: Use integration by parts to show that < 0o, if y itself is a normalized solution to the Schrödinger | (a-v)*(a_v) dx = v*(a,a_v)dx. 00 Then invoke the Schrödinger equation (Equation 2.46) to obtain 1. la-v* dx = E -ħw, 2 where E is the energy of the state y.
Problem 2.11 Show that the lowering operator cannot generate a state of infinite norm (i.e., f la.-²dx < equation). What does this tell you in the case y = vo? Hint: Use integration by parts to show that < 0o, if y itself is a normalized solution to the Schrödinger | (a-v)*(a_v) dx = v*(a,a_v)dx. 00 Then invoke the Schrödinger equation (Equation 2.46) to obtain 1. la-v* dx = E -ħw, 2 where E is the energy of the state y.
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![(a_)*(a_V)ax- J-o
Problem 2.11 Show that the lowering operator cannot generate a state of infinite
norm (i.e., f la.-²dx < oo, if y itself is a normalized solution to the Schrödinger
equation). What does this tell you in the case y = vo? Hint: Use integration by
parts to show that
y*(a,a_) dx.
=
-00
Then invoke the Schrödinger equation (Equation 2.46) to obtain
la-yl² dx E - hw,
-0-
where E is the energy of the state y.
**Problem 2.12
(a) The raising and lowering operators generate new solutions to the Schrödinger
equation, but these new solutions are not correctly normalized. Thus a Vn
is proportional to yn+1, and a n is proportional to yn-1, but we'd like to
know the precise proportionality constants. Use integration by parts and the
Schrödinger equation (Equations 2.43 and 2.46) to show that
roo
| la+ Vl² dx = (n+ 1)hw,
la- Vnl? dx = nhw,
-00
-00
and hence (with i's to keep the wavefunctions real)
a+ Vn = iv(n + 1)hw yn+1,
[2.52]
a_n = -ivnhw n-1.
[2.53]
Sec. 2.3: The Harmonic Oscillator
37
(b) Use Equation 2.52 to determine the normalization constant A, in Equation 2.50.
(You'll have to normalize o "by hand".) Answer:
1/4
(-i)"
An =
[2.54]
Vn!(hw)"
*Problem 2.13 Using the methods and results of this section,
(a) Normalize n (Equation 2.51) by direct integration. Check your answer against
the general formula (Equation 2.54).
(b) Find 2, but don't bother to normalize it.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61b99ad3-eb7b-4da5-a575-35a21eff0048%2Fb8c0efe5-5e63-4369-a058-2b270e2baa2c%2Funh0iv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a_)*(a_V)ax- J-o
Problem 2.11 Show that the lowering operator cannot generate a state of infinite
norm (i.e., f la.-²dx < oo, if y itself is a normalized solution to the Schrödinger
equation). What does this tell you in the case y = vo? Hint: Use integration by
parts to show that
y*(a,a_) dx.
=
-00
Then invoke the Schrödinger equation (Equation 2.46) to obtain
la-yl² dx E - hw,
-0-
where E is the energy of the state y.
**Problem 2.12
(a) The raising and lowering operators generate new solutions to the Schrödinger
equation, but these new solutions are not correctly normalized. Thus a Vn
is proportional to yn+1, and a n is proportional to yn-1, but we'd like to
know the precise proportionality constants. Use integration by parts and the
Schrödinger equation (Equations 2.43 and 2.46) to show that
roo
| la+ Vl² dx = (n+ 1)hw,
la- Vnl? dx = nhw,
-00
-00
and hence (with i's to keep the wavefunctions real)
a+ Vn = iv(n + 1)hw yn+1,
[2.52]
a_n = -ivnhw n-1.
[2.53]
Sec. 2.3: The Harmonic Oscillator
37
(b) Use Equation 2.52 to determine the normalization constant A, in Equation 2.50.
(You'll have to normalize o "by hand".) Answer:
1/4
(-i)"
An =
[2.54]
Vn!(hw)"
*Problem 2.13 Using the methods and results of this section,
(a) Normalize n (Equation 2.51) by direct integration. Check your answer against
the general formula (Equation 2.54).
(b) Find 2, but don't bother to normalize it.
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