3. We discussed a perturbative expansion for solving the Schrodinger equation for scat- tering of a particle of energy E = ħ²k²/2m where the first non-trivial approximation (Born approximation) was vy (x) 2m ikx + G(x-x')V (x')eika' dx' with G(xx) the Green's function we discussed in class. Use this to compute the reflection and transmission coefficients as a function of k for scattering off the potential V(x) = Voe-αx² where Vo and a are positive, real constants. 4. Let A, B be operators and as usual define the exponential of an operator by e Σo An/n!. In this problem you will derive some results which are very helpful in dealing with exponentials of operators. These results will be particularly helpful when we discuss the quantum simple harmonic oscillator. They generally go under the name of Baker Campbell Hausdorff or BCH formula. (a) We want to prove a relation expressing eA Be-A as an infinite sum of nested commutators: eA Be-A = 1 1 : B + [A, B] + [A, [A, B}]] + ¸[A, [A, [A, B}]] + … … + + ... To prove this introduce a real parameter A and consider the operator valued function of X f(A)=e^A Be-A By differentiating with respect to A derive a first order differential equation satisfied by f(A). From this equation and the initial condition obeyed by f(0), derive the solution to this first order differential equation as a power series in A. Finally, set = 1 to derive the desired result. Be very careful with the order of operators in this problem. (b) Suppose now that A, B are operators that commute with their commutator, that is such that [A, [A, B]] = [B, [A, B] = 0. Show that [A", B]=nA1A, B] for all positive integers n = 1, 2, 3,.... (c) Assuming still that A, B are operators that commute with their commutator, show that eAB=BAA,B]
3. We discussed a perturbative expansion for solving the Schrodinger equation for scat- tering of a particle of energy E = ħ²k²/2m where the first non-trivial approximation (Born approximation) was vy (x) 2m ikx + G(x-x')V (x')eika' dx' with G(xx) the Green's function we discussed in class. Use this to compute the reflection and transmission coefficients as a function of k for scattering off the potential V(x) = Voe-αx² where Vo and a are positive, real constants. 4. Let A, B be operators and as usual define the exponential of an operator by e Σo An/n!. In this problem you will derive some results which are very helpful in dealing with exponentials of operators. These results will be particularly helpful when we discuss the quantum simple harmonic oscillator. They generally go under the name of Baker Campbell Hausdorff or BCH formula. (a) We want to prove a relation expressing eA Be-A as an infinite sum of nested commutators: eA Be-A = 1 1 : B + [A, B] + [A, [A, B}]] + ¸[A, [A, [A, B}]] + … … + + ... To prove this introduce a real parameter A and consider the operator valued function of X f(A)=e^A Be-A By differentiating with respect to A derive a first order differential equation satisfied by f(A). From this equation and the initial condition obeyed by f(0), derive the solution to this first order differential equation as a power series in A. Finally, set = 1 to derive the desired result. Be very careful with the order of operators in this problem. (b) Suppose now that A, B are operators that commute with their commutator, that is such that [A, [A, B]] = [B, [A, B] = 0. Show that [A", B]=nA1A, B] for all positive integers n = 1, 2, 3,.... (c) Assuming still that A, B are operators that commute with their commutator, show that eAB=BAA,B]
Principles of Physics: A Calculus-Based Text
5th Edition
ISBN:9781133104261
Author:Raymond A. Serway, John W. Jewett
Publisher:Raymond A. Serway, John W. Jewett
Chapter28: Quantum Physics
Section: Chapter Questions
Problem 25P
Related questions
Question
![3. We discussed a perturbative expansion for solving the Schrodinger equation for scat-
tering of a particle of energy E = ħ²k²/2m where the first non-trivial approximation
(Born approximation) was
vy (x)
2m
ikx
+
G(x-x')V (x')eika' dx'
with G(xx) the Green's function we discussed in class. Use this to compute
the reflection and transmission coefficients as a function of k for scattering off the
potential
V(x) = Voe-αx²
where Vo and a are positive, real constants.
4. Let A, B be operators and as usual define the exponential of an operator by e
Σo An/n!. In this problem you will derive some results which are very helpful
in dealing with exponentials of operators. These results will be particularly helpful
when we discuss the quantum simple harmonic oscillator. They generally go under
the name of Baker Campbell Hausdorff or BCH formula.
(a) We want to prove a relation expressing eA Be-A as an infinite sum of nested
commutators:
eA Be-A
=
1
1
: B + [A, B] + [A, [A, B}]] + ¸[A, [A, [A, B}]] + … …
+ + ...
To prove this introduce a real parameter A and consider the operator valued
function of X
f(A)=e^A Be-A
By differentiating with respect to A derive a first order differential equation
satisfied by f(A). From this equation and the initial condition obeyed by f(0),
derive the solution to this first order differential equation as a power series in A.
Finally, set = 1 to derive the desired result. Be very careful with the order of
operators in this problem.
(b) Suppose now that A, B are operators that commute with their commutator, that
is such that [A, [A, B]] = [B, [A, B] = 0. Show that
[A", B]=nA1A, B]
for all positive integers n = 1, 2, 3,....
(c) Assuming still that A, B are operators that commute with their commutator,
show that
eAB=BAA,B]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc93fdf08-f87f-4600-aa7f-2f43250f8ca7%2F10a5d338-6427-4b5a-82ea-350ccf2b7806%2Fjaqj2k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. We discussed a perturbative expansion for solving the Schrodinger equation for scat-
tering of a particle of energy E = ħ²k²/2m where the first non-trivial approximation
(Born approximation) was
vy (x)
2m
ikx
+
G(x-x')V (x')eika' dx'
with G(xx) the Green's function we discussed in class. Use this to compute
the reflection and transmission coefficients as a function of k for scattering off the
potential
V(x) = Voe-αx²
where Vo and a are positive, real constants.
4. Let A, B be operators and as usual define the exponential of an operator by e
Σo An/n!. In this problem you will derive some results which are very helpful
in dealing with exponentials of operators. These results will be particularly helpful
when we discuss the quantum simple harmonic oscillator. They generally go under
the name of Baker Campbell Hausdorff or BCH formula.
(a) We want to prove a relation expressing eA Be-A as an infinite sum of nested
commutators:
eA Be-A
=
1
1
: B + [A, B] + [A, [A, B}]] + ¸[A, [A, [A, B}]] + … …
+ + ...
To prove this introduce a real parameter A and consider the operator valued
function of X
f(A)=e^A Be-A
By differentiating with respect to A derive a first order differential equation
satisfied by f(A). From this equation and the initial condition obeyed by f(0),
derive the solution to this first order differential equation as a power series in A.
Finally, set = 1 to derive the desired result. Be very careful with the order of
operators in this problem.
(b) Suppose now that A, B are operators that commute with their commutator, that
is such that [A, [A, B]] = [B, [A, B] = 0. Show that
[A", B]=nA1A, B]
for all positive integers n = 1, 2, 3,....
(c) Assuming still that A, B are operators that commute with their commutator,
show that
eAB=BAA,B]
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