Problems and solutions to practice 1 - 11 PDF

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PHYSICS 621 - Fall Semester 2012 - ODU

Graduate Quantum Mechanics - Problem Set 1 Problem 1) Write down the total mechanical energy (kinetic plus potential) of a mass m in free fall, expressing it in terms of the momentum p and the height x above ground: E = Tkin +V (x) = H ( p, x) . Take the partial

∂H = v = x . Then ∂p show that the negative of the partial derivative with respect to x equals the force, i.e. the rate of change ∂H = F = p of the momentum: − ∂x derivative of the function H with respect to p and show that it is equal to the velocity,

Problem 2)  r Show that a vector potential given (in cylindrical coordinates) by A(r⊥,ϕ, z) = b ϕˆ corresponds to a 2    ˆ A = b B = ∇ × z along the z-axis. (You may use the formula sheet) constant magnetic field

Problem 3) What is the force of a magnetic field of 0.1 Tesla in z-direction on an electron instantaneously moving along the x-axis with 10% of the speed of light? What is the equation of motion for this electron? What kind of motion does it describe (give all the numeric parameters, e.g. amplitude and period if the motion is periodic).

Problem 4) For each of the following statements about Quantum Mechanics, indicate whether you believe them to be correct or wrong. Give a 1-2 sentence explanation for each of your responses: a. If all possible information on a system is given, Quantum Mechanics can predict the outcome of any future measurement on the system accurately. b. Quantum Mechanics cannot predict anything precisely c. Quantum Mechanics cannot predict with certainty the result of any particular measurement on a single particle d. The Heisenberg Uncertainty principle means that nothing can be measured precisely e. The x- and y- components of any angular momentum cannot simultaneously be measured with arbitrary precision. f. The time evolution of a quantum mechanical wave function is described by a unitary operator.

PHYSICS 621 - Fall Semester 2012 - ODU

Problem 5) d 2 y(x) − m 2 y(x) = 0 for real m. Make sure you find the most general dx 2 solution – what are the “integration constants”?

Solve the differential equation

Problem 6) Proof that for any complex numbers c, z with z = exp ( c) we have z* = exp( c *)

Problem 7) Find the Fourier transform f (p) of the function 2 1 exp − x 2 : 2π ∞ f ( p) = 1 ∫ f (x)exp(−ipx) dx 2π −∞

f (x) =

(

)

PHYSICS 621 - Fall Semester 2012 - ODU

Graduate Quantum Mechanics - Problem Set 1 - Solutions Problem 1) E = Tkin +V (x) = H ( p, x) =

p2 + mgx . 2m

∂H p = = v = x ∂p m −

∂H = −mg = F = p ∂x

Problem 2)  r r A(r⊥,ϕ , z) = b ϕˆ ⇒ Aϕ = b, Ar = 0, Az = 0 ⇒ 2 2    ∂A 1 ∂rAϕ 1 ˆz = rbˆz = b zˆ B = ∇ × A = − ϕ rˆ + ∂z r ∂r r where we took only the non-zero terms from the curl in cylindrical coordinates in the formula sheet. This leads to motion in a circle of radius

Problem 3)    ˆ = 3⋅10 7e N/C yˆ = 4.8 ⋅10 −13 N yˆ F = q v × B = −e (0.1c ) (0.1T ) (− y)

The magnitude of this force is constant and it is always perpendicular to the direction of motion. mv Therefore, the electron moves on a circle of radius R = = 0.0017 m centered at x = 0, y = R. Its eB v angular velocity is ω = = 1.76 ⋅1010 rad/s which corresponds to a full orbit every 0.36 ns. R

Problem 4) For each of the following statements about Quantum Mechanics, indicate whether you believe them to be correct or wrong. Give a 1-2 sentence explanation for each of your responses: a. If all possible information on a system is given, Quantum Mechanics can predict the outcome of any future measurement on the system accurately. WRONG: in general, only probabilities can be predicted b. Quantum Mechanics cannot predict anything precisely. WRONG – then it wouldn’t be physics!

PHYSICS 621 - Fall Semester 2012 - ODU c. Quantum Mechanics cannot predict with certainty the result of any particular measurement on a single particle. DEPENDS – if a particle is in an eigenstate of an observable, I can predict the outcome of a measurement of that observable precisely. d. The Heisenberg Uncertainty principle means that nothing can be measured precisely. WRONG – see above e. The x- and y- components of any angular momentum cannot simultaneously be measured with arbitrary precision. CORRECT f. The time evolution of a quantum mechanical wave function is described by a unitary operator. CORRECT

Problem 5) The most general solution is y(x) = A exp(mx) + B exp(−mx) which can be shown by plugging it in (as a 2nd order differential equation, there must be two integration constants, A and B). Since y(0)=A+B and y’(0)=mA-mB, we can solve for A and B in terms of the initial conditions at x=0.

Problem 6) z = exp(c) = exp(Re(c) + i Im(c)) = exp(Re(c)) (cos(Im(c)) + isin(Im(c)) ) z * = exp(Re(c)) (cos(Im(c)) − isin(Im(c)) ) = exp(Re(c))exp(−i Im(c)) = exp(c*)

Problem 7) See next recitation

PHYSICS 621 - Fall Semester 2013 - ODU

Graduate Quantum Mechanics - Problem Set 2 Problem 1) Do continuous functions defined on the interval [0…L] and that vanish at the end points x = 0 and x = L form a vector space? How about periodic functions obeying f(L) = f(0)? How about all functions with f(0)=4? If the functions do not qualify, list the things that go wrong.

Problem 2) Consider the vector space V spanned by real 2x2 matrices. What is its dimension? What would be a suitable basis? Consider three example “vectors” from this space:

! $ 1 = # 0 1 &; " 0 0 %

! 1 1 $ 2 =# &; " 0 1 %

! −2 −1 $ 3 =# & " 0 −2 %

Are they linearly independent? Support your answer with details.

Problem 3)   Consider the two vectors A = 3iˆ + 4 ˆj and B = 2iˆ − 6 ˆj in the 2-dimensional space of the x-y plane. Do they form a suitable set of basis vectors? (Explain.) Do they form an orthonormal basis set? If not, follow the construction in the book (p.15) and the lecture to turn them into an othomormal set.

Problem 4) Prove the triangle inequality V + W ≤ V + W

for arbitrary vectors in any vector space with an

inner product. You may use the Schwarz Inequality V W ≤ V ⋅ W .

Problem 5) Assume the two operators Ω and Λ are Hermitian. What can you say about i) ii) iii) iv)

ΩΛ; ΩΛ+ ΛΩ; [Ω , Λ] = Ω Λ − Λ Ω; i [Ω , Λ ] ?

Problem 6) ! cos ϕ isin ϕ Prove that the matrix # #" isin ϕ cosϕ

$ & is unitary. &%

PHYSICS 621 - Fall Semester 2013 - ODU

Graduate Quantum Mechanics - Problem Set 2 - Solutions Problem 1) a) Do functions defined on the interval [0…L] and that vanish at the end points x = 0 and x = L form a vector space? Answ.: Yes. This is just a special case of the one discussed in lectures; clearly f(0) = f(L) here. b) How about periodic functions obeying f(L) = f(0)? Answ.: Yes. This is a generalization of the case discussed in lecture. Periodic functions form a subspace of all functions on the real numbers. (It may be impossible, though, to introduce a workable inner product). c) How about all functions with f(0)=4? Answ.: No. This “vector space” wouldn’t behave properly under addition: (f+g)(x) = f(x) + g(x) wouldn’t work for x = 0 (4+4 ≠ 4).

Problem 2) Consider the vector space V spanned by real 2x2 matrices. What is its dimension? What would be a suitable basis? Consider three example “vectors” from this space:

! $ 1 = # 0 1 &; " 0 0 %

! 1 1 $ 2 =# &; " 0 1 %

! −2 −1 $ 3 =# & " 0 −2 %

Are they linearly independent? Support your answer with details. Answ: Similar as the case we discussed in lecture, this vector space has 4 dimensions. A simple basis could be

! $ b1 = # 1 0 & ; " 0 0 %

! $ b2 = # 0 1 & ; " 0 0 %

! $ b3 = # 0 0 &; " 1 0 %

! $ b4 = # 0 0 & " 0 1 %

The three vectors given are not linearly independent, since

% " 2 2 % " −2 −1 % " 1 − 2 2 = $ 0 1 ' −$ ' '= 3 =$ # 0 −2 & # 0 0 & # 0 2 &

Problem 3)   Consider the two vectors A = 3iˆ + 4 ˆj and B = 2iˆ − 6 ˆj in the 2-dimensional space of the x-y plane. Do they form a suitable set of basis vectors? (Explain.) Do they form an orthonormal basis set? If not, follow the construction in the book (p.15) and the lecture to turn them into an othomormal set.

PHYSICS 621 - Fall Semester 2013 - ODU

Answ.: The 2 vectors are linearly independent (you can’t write one as a multiple of the other). Since the space has only 2 dimensions, they therefore form a basis. However, they are neither normalized nor orthogonal to each other. To turn them into an orthonormal set, first we have to normalize the first one:  3iˆ + 4 ˆj A = 0.6iˆ + 0.8 jˆ . Aˆ =  = A 32 + 4 2 Then, we determine the orthogonal part of the 2nd vector:    ˆ Aˆ = (2iˆ − 6 ˆj) − (1.2 − 4.8)(0.6i ˆ+ 0.8 j) ˆ = 4.16 iˆ − 3.12 jˆ B⊥ = B − ( B ⋅ A) As the final step, we have to normalize this vector:  ˆ ˆ ˆB⊥ = B⊥ = 4.16 i − 3.12 j = 0.8iˆ − 0.6 ˆj 5.2 B⊥

Aˆ and Bˆ⊥ form an orthonormal basis.

Problem 4) for arbitrary vectors in any vector space with an

Prove the triangle inequality V + W ≤ V + W

inner product. You may use the Schwarz Inequality V W ≤ V ⋅ W . Answ.: Since both sides are clearly positive, it is sufficient to show V +W

2

(

≤ V + W

)

2

⇔ V +W V +W ≤ V

2

+ W

V V + W W + V W + W V ≤ V V + W W +2 V 2 Re( V W

) ≤2 V

2

+2 V

W ⇔

W ⇔

W

The last line follows from the Schwarz Inequality since the real part of any complex number is less or equal to its absolute value.

Problem 5) Assume the two operators Ω and Λ are Hermitian. What can you say about i)

ΩΛ; Answ.: The product is not necessarily Hermitian since the 2 operators don’t necessarily commute: (Ω Λ)† = Λ† Ω† = Λ Ω ≠ Ω Λ

PHYSICS 621 - Fall Semester 2013 - ODU

ii) iii) iv)

ΩΛ+ ΛΩ; Answ.: This is Hermitian: (Ω Λ + Λ Ω)† = Λ† Ω† + Ω† Λ† = Λ Ω + Ω Λ = Ω Λ + Λ Ω [Ω , Λ] ; Answ.: This is not Hermitian (unless the commutator is zero). Instead, [Ω , Λ]† = − [Ω , Λ] i [Ω , Λ ] ? Answ.: This is Hermitian: (i [Ω , Λ])† = -i [Ω , Λ]† = -i (− [Ω , Λ]) = −i [Ω , Λ]

Problem 6) ! cos ϕ isin ϕ $ & is unitary. Prove that the matrix # #" isin ϕ cosϕ &% " cos ϕ −isin ϕ % ' (interchanging columns and rows and taking the Answ.: The adjoint matrix is $ $# −isinϕ cosϕ '& complex conjugate). The product between these is ! cos ϕ isin ϕ $! cos ϕ −isinϕ $ ! cos2 ϕ + sin 2 ϕ −i cos ϕ sin ϕ + isin ϕ cosϕ $& ! 1 0 $ # &# &=# =# & & " 0 1 % #" isin ϕ cosϕ &%#" −isin ϕ cosϕ &% #" isin ϕ cosϕ − i cos ϕ sin ϕ sin 2 ϕ + cos2 ϕ %

q.e.d.

PHYSICS 621 - Fall Semester 2013 - ODU

Graduate Quantum Mechanics - Problem Set 3 Problem 1) " 0 0 1 % $ ' Consider the matrix Ω = $ 0 0 0 ' . $# 1 0 0 '& i) ii) iii) iv)

Is it hermitian? Find its eigenvalues and eigenvectors Verify that U†ΩU is diagonal, U being the matrix formed by using each normalized eigenvector as one of its columns. (Show that U is unitary!) ∞ 1 n Calculate the matrix exp ( iΩ) = ∑ ( iΩ) (where, for any matrix, M0 = 1). Show that it is n=0 n! unitary.

Problem 2) Show that δ(ax − b) =

b 1 δ (x − ) by evaluating a a

∞

∫ f (x) δ(ax − b)dx

for an arbitrary function f(x).

−∞

Consider the two cases a > 0 and a < 0 separately.

Problem 3) # 1 if x ≥ x ' dθ (x − x ') Consider the “Theta-funcion” θ (x − x ') = $ . Show that δ(x − x ') = by dx % 0 else multiplying both the l.h.s. and the r.h.s. with an arbitrary square-integrable function f(x) and integrating over all x.

Problem 4) A stone dropping from a height h = 5m follows a path y(t) = h – ½ gt2 (g = 10 m/s2) until it hits the ground after exactly 1 second. Now consider a more general path y(t) = h – ½ gt2 + a sin(πt) with arbitrary constant a (and t measured in seconds). Obviously, this path has the same start and end points (at the same times) as the correct one. Show that the action (the integral over the proper Lagrangian for this situation along the second path) between these two endpoints has an extremum for the case a = 0. [This is a very specific test of the general statement about the “principle of least action” in the lecture. You must do the math explicitly; don’t just quote the general principle.]

PHYSICS 621 - Fall Semester 2013 - ODU

Graduate Quantum Mechanics - Problem Set 3 - Solution Problem 1) " 0 0 1 % $ ' Consider the matrix Ω =$ 0 0 0 ' . $# 1 0 0 '& i) ii)

iii)

Is it hermitian? Answ.: Yes – it is identical to its adjoint (swapping rows and columns). Find its eigenvalues and eigenvectors Answ.: The eigenvalues are +1, 0, -1 (solutions to the characteristic equation λ3 – λ = 0). ! 1 $ ! 1 $ & # & # 2 & # 2 & !# 0 $& # The corresponding normalized eigenvectors are # 0 & , # 1 & , # 0 & . & # & # # 1 & #" 0 &% # − 1 & # # 2 & 2 & % " % " † Verify that U ΩU is diagonal, U being the matrix formed by using each normalized eigenvector as one of its columns. (Show that U is unitary!) Answ.: " 1 " 1 1 % 1 % ' ' $ $ 0 0 2 ' 2 ' $ 2 $ 2 0 ', U† = $ 0 1 0 ' = U, U =$ 0 1 ' ' $ $ $ 1 0 − 1 ' $ 1 0 − 1 ' $ 2 $ 2 2 ' 2 ' & & # # q.e.d % %" 1 " 1 1 1 ' '$ $ 0 − 0 2 ' " 1 0 0 % 2 '$ 2 $ 2 ' $ 0 '=$ 0 0 0 ' 0 '$ 0 0 U †ΩU = $ 0 1 ' '$ $ 1 ' $# 0 0 −1 '& $ 1 0 − 1 '$ 1 0 $ 2 2 ' 2 '$ 2 & &# # ∞

iv)

1 n ( iΩ) (where, for any matrix, M0 = 1). Show that it is unitary. n=0 n!

Calculate exp ( iΩ) = ∑ Answ.:

" 1 0 0 % " 0 0 i % " −1 0 0 $ ' ' $ 1 $ 2 (iΩ) = $ 0 1 0 ' , (iΩ ) = $ 0 0 0 ' , ( iΩ) = $ 0 0 0 $ 0 0 1 ' $ i 0 0 ' $ 0 0 −1 # # & & # 0

and from there on, it repeats. Collecting all terms, we find

" 0 0 −i % % $ ' ' 3 ' , ( iΩ) = $ 0 0 0 ' = −iΩ $ −i 0 0 ' ' # & &

PHYSICS 621 - Fall Semester 2013 - ODU

# 1− 1 + 1 −... 0 i(1− 16 + 1201 −...) 24 2 % exp(iΩ) = % 0 1 0 %% i(1− 1 + 1 −...) 0 1− 1 + 1 −... 120 24 2 6 $ which clearly is a unitary matrix.

& # & ( % cos(1) 0 isin(1) ( ( =% ( 0 1 0 (( % isin(1) 0 cos(1) ( ' ' $

Problem 2) Show that δ(ax − b) =

b 1 δ (x − ) by evaluating a a

∞

∫ f (x) δ(ax − b)dx

for an arbitrary function f(x).

−∞

Consider the two cases a > 0 and a < 0 separately. Answ.: We show that we get the same result after integration for both forms: ∞

∫

! ∞

f (x) δ(ax − b)dx =

−∞

∫

f (x)

−∞

b 1 1 b δ(x − )dx = f ( ) a a a a

where we integrated the r.h.s. following the standard rules for the delta-function. For the left hand side, we make a variable substitution: u = ax, du = adx. Therefore, if a >0, ∞ a⋅∞ 1 b ∫ f (x) δ(ax − b)dx = 1 / a ∫ f ( ua )δ (u − b)du = a f (a ) where we use a∞ = ∞ and |a| = a for positive a, −∞ −a⋅∞ q.e.d. For negative a, the only change is that the factor 1/a in front of the integral becomes negative and therefore equal to -1/|a|. However, the limits of integration are multiplied with a negative number at the same time, so that the integral over u would now go from +∞ to -∞. Changing the order back by interchanging upper and lower integration limit yields another minus sign which cancels the first one, yielding the desired result.

Problem 3) # 1 if x ≥ x ' dθ (x − x ') Consider the “Theta-funcion” θ (x − x ') = $ . Show that δ(x − x ') = by dx % 0 else multiplying both the l.h.s. and the r.h.s. with an arbitrary square-integrable function f(x) and integrating over all x. Answ.: We show that we get the same result after integration of both sides: ∞

∫ f (x) δ(x − x ') dx = f (x ') −∞ ∞

∫ f (x) −∞

∞ d θ (x − x ') ∞ dx = [ f (x)θ (x − x ')] −∞ − ∫ f '(x) θ (x − x ') dx = 0 − dx −∞

∞

∫ f '(x) dx = −[ f (x)]

∞ x'

= f (x ')

x'

where the first line follows from the definition of the delta function. In the 2nd line, we make use of the fact that any square-integrable function f(x) must converge to zero at ±∞, and the definition θ(x-x’).

PHYSICS 621 - Fall Semester 2013 - ODU

Problem 4) A stone dropping from a height h = 5m follows a path y(t) = h – ½ gt2 (g = 10 m/s2) until it hits the ground after exactly 1 second. Now consider a more general path y(t) = h – ½ gt2 + a sin(πt) with arbitrary constant a (and t measured in seconds). Obviously, this path has the same start and end points (at the same times) as the correct one. Show that the action (the integral over the proper Lagrangian for this situation along the second path) between these two endpoints has an extremum for the case a = 0. [This is a very specific test of the general statement about the “principle of least action” in the lecture. You must do the math explicitly; don’t just quote the general principle.] Answ.: The Lagrangian in this case is given by m L = T −V = y 2 − mgy . Plugging in the proposed form of y(t), we find 2 2 1 y = h − 2 gt + asin π t , y = −gt + a π cos πt

m 22 2 2 ( g t + a π cos2 πt − 2gt a π cosπ t) − mg (h − 12 gt 2 + asin π t) 2 m = mg 2 t 2 + a 2π 2 cos2 πt − mgt aπ cosπt − mgasin πt − mgh 2 Integrating each term over the interval 0 ≤ t ≤ 1, we find for the action S: 1 1 13 m 1 S = mg 2 + a 2π 2 − mgt asin π t 0 + mga ∫ sin π tdt − mga ∫ sin πt dt − mgh1 = 3 4 0 0 L(t) =

mg 2 m 2 2 + a π − mgh 3 4 Clearly, S has an extremum (its derivative with respect to a is zero) for a = 0, q.e.d =

PHYSICS 621 - Fall Semester 2013 - ODU

Graduate Quantum Mechanics - Problem Set 4 Problem 1) Assuming a particle is described with the usual cartesian coordinates (x,y,z) and momenta (px, py, pz). Write down the x, y and z components of the angular momentum operator in terms of these canonical variables. Calculate the Poisson brackets {Lx,Lz} and {Ly,Lz} explicitely. Given our interpretation of Lz as “generator” of rotations around the z axis, can you interpret your result in terms of the transformation of the vector L under the coordinate transformation generated by Lz?

Problem 2) Write down the Lagrangian for two equal masses m at positions x1 and x2 (each measured relative to the equilibrium position), coupled to each other and (on their other sides) to two fixed walls with springs with constant k but otherwise free to move along the x-direction. If the system is in equilibrium, all three springs are relaxed (unstretched/compressed). [This is exactly the set up in Example 1.8.6 in Shankar’s book, p. 46.] Set up the Lagrangian for this system. Find the generalized momenta. Then follow the explicit procedure (Legendre transformation) in the lecture to find the corresponding Hamiltonian. Write down Hamilton’s canonical equations. (You don’t have to solve them).

Problem 3) Use the vector potential representation of a constant magnetic field B along the z-axis from our first homework problem set. The Hamiltonian for this case in cylindrical coordinates (r⊥, ϕ, z) with canonical momenta (Pr⊥ , Pϕ , Pz ) is given by 2

# Pϕ & 2 2 % r − qAφ ( + Pr⊥ + Pz ' H=$ ⊥ 2m By writing down Hamilton’s equations of motion, give an interpretation (in terms of the “usual” momenta or velocities) of (Pr⊥ , Pϕ , Pz ). Which of these three are conserved? Under what condition are all 3 conserved? Can you interpret this condition? [What does it mean physically?]

PHYSICS 621 - Fall Semester 2013 - ODU

Graduate Quantum Mechanics - Problem Set 4 - Solution Problem 1) Assuming a particle is described with the usual cart...