Fakultat fur Physik

R: Rechenmethoden fur Physiker, WiSe 2015/16

Dozent: Jan von Delft

Ubungen: Benedikt Bruognolo, Dennis Schimmel,

Frauke Schwarz, Lukas Weidinger

http://homepages.physik.uni-muenchen.de/~vondelft/Lehre/15r/

Sheet 06.6: Curvilinear Integration, Matrices I

Posted: Friday, 20.11.15 Due: Friday, 27.11.15, 13:00 Central Tutorial: 02.12.15[2](E/M/A) means: problem counts 2 points and is easy/medium hard/advanced

Example Problem 1: Exponential integrals of the form´∞0

dx xne−ax [2]Points: (a)[1](M); (b)[1](M)

Calculate the integral In(a) =´∞0

dx xne−ax (with a ∈ R, a > 0, n ∈ N) using two differentmethods: (a) repeated partial integration, and (b) repeated differentiation:

(a) Calculate I0, I1 and I2 by using partial integration where necessary. Then use partial integrationto show that In(a) = n

aIn−1(a) for all n ≥ 1. Use this relation iteratively to determine In(a)

as a function of a and n.

(b) Show that taking n derivatives of I0(a) with respect to a yields In(a) = (−1)n dnI0(a)dan

. Thencalculate these derivatives for a few small values of n. From the emerging pattern, deduce thegeneral formula for In(a).

Example Problem 2: Elliptical polar coordinates: area of an ellipse [2]Points: (a)[1](M); (b)[1](E)

(a) Let f : R2 → R be a function that depends on the coordinates x and y only in the combination(x/a)2 + (y/b)2. Show that a two-dimensional area integral of f over R2 can be written as

I =

ˆR2

dxdy f((x/a)2 + (y/b)2

)= 2πab

ˆ ∞0

dµµ f(µ) ,

by transforming from cartesian coordinates to elliptical polar coordinates, defined as follows:

x = µa cosφ, y = µb sinφ ,

µ2 = (x/a)2 + (y/b)2, φ = arctan(ay/bx) .

Hint: For a = b = 1, they correspond to polar coordinates. For a 6= b, the local basis is notorthogonal!

(b) Using a suitable function f , calculate the area of an ellipse with semi-axes a and b, with aand b defined by (x/a)2 + (y/b)2 ≤ 1.

Example Problem 3: Jacobian determinant for cylindrical coordinates [1]Points: [1](E)

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Calculate the Jacobian determinant∣∣∣∂(x1,x2,x3)∂(ρ,φ,z)

∣∣∣ for the transformation from Cartesian to cylindrical

coordinates.

Example Problem 4: Volume and moment of inertia of a conical frustum [2]Points: (a)[1](E); (b)[1](E)

The moment of inertia of a rigid body with respect to a rotation axis is defined as I =´V

dV ρ0(r)d2⊥(r),

where ρ0(r) is the density at the point r, and d⊥(r) the perpendicular distance from r to the ro-tation axis.

Let F = {r ∈ R3 |H ≤ z ≤ 2H,√x2 + y2 ≤ az} be a homogeneous conical

frustum centered on the z-axis. Calculate, using cylindrical coordinates,

(a) its volume VF (a), and

(b) its moment of inertia IF (a) with respect to the z axis,

ze

H

H2

as functions of the dimensionless, positive scale factor a, the length parameter H, and the massm of the frustum. [Check your results: VF (3) = 21πH3, IF (1) = 93π

70MH2.]

Example Problem 5: Volume of a buoy [2]Points: (a)[1](E); (b)[1](M)

Consider a buoy, with its tip at the origin, bounded from above by a sphere centeredthe origin, with x2 + y2 + z2 ≤ R2, and from below by a cone with tip at the origin,with z ≥ a

√(x2 + y2). θ

z

R

(a) Show that the half angle at the tip of the cone is given by θ = arctan(1/a).

(b) Use spherical coordinates to calculate the volume V (R, a) of the buoy as a function of R anda. [Check your results: V (2,

√3) = (16π/3)(1−

√3/2).]

Example Problem 6: Wave functions of the two-dimensional harmonic oscillator [4]Points: (a)[0,5](E); (b)[0,5](E); (c)[3](M)

The quantum mechanical treatment of a two-dimensional harmonic oscillator leads to so-called‘wave functions’,

Ψnm : R2 → C, r 7→ Ψnm(r) , with n ∈ N0, m ∈ Z, m = −n,−n+ 2, . . . , n− 2, n,

which have a factorized form when written in terms of polar coordinates, Ψnm(r) = Rn|m|(ρ)Zm(φ),with Zm(φ) = 1√

2πeimφ. The wave functions satisfy the following ‘orthogonality relation’:

Omm′

nn′ ≡ˆR2

dAΨnm(r)Ψn′m′(r) = δnn′δmm′ .

Verify these for n = 0, 1 and 2, where the radial wave functions have the form:

R00(ρ) =√

2e−ρ2/2, R11(ρ) =

√2ρe−ρ

2/2, R22(ρ) = ρ2 e−ρ2/2, R20(ρ) =

√2[ρ2 − 1]e−ρ

2/2.

Proceed as follows. Due to the product form of the wave function Ψ, each area integral separates

into two factors that can be calculated separately, Omm′

nn′ = P|m||m′|nn′ Pmm′

, where P is a radial

integral and P an angular integral.

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(a) Find general expressions for P and P as integrals over R or Z functions, respectively.

(b) Compute the angular integral Pmm′for arbitrary values of m and m′.

(c) Now compute those radial integrals that arise in combination with P 6= 0, namely P 0000 , P 11

11 ,P 2222 , P 00

22 and P 0020 .

Hint: The Euler identity, ei2πk = 1 if k ∈ Z, is useful for evaluating the angular integral, and´∞0

dx xne−x = n! for the radial integrals.

Background information: The functions Ψnm(r) are the ‘eigenfunctions’ of a quantum mechani-cal particle in a two-dimensional harmonic potential, V (r) ∝ r2, where n and m are ‘quantumnumbers’ that specify a particular ‘eigenstate’. A particle in this state is found with probability|Ψnm(r)|2dA within the area element dA at position r. The total probability for being found any-where in R2 equals 1, hence the normalization integral yields Omm

nn = 1 for every eigenfunctionΨnm(r). The fact that the area integral of two eigenfunctions vanishes if their quantum numbersare not equal, reflects the fact that the eigenfunctions form an orthonormal basis in the space ofsquare-integrable complex functions on R2.

Example Problem 7: Matrix multiplication [2]Points: [2](E)

Compute all possible products of two of the following matrices:

P =(

4 −3 1

2 2 −4

), Q =

(3 0 1

1 2 5

1 −6 −1

), R =

(3 0

1 2

1 −6

).

Example Problem 8: Spin 12matrices [3]

Points: (a)[0,5](E); (b)[1,5](E); (c)[1](E)

The following matrices are used for the description for quantum mechanical particles with spin 12:

Sx = 12

(0 11 0

), Sy = 1

2

(0 −ii 0

), Sz = 1

2

(1 00 −1

).

(a) Compute S2 = S2x + S2

y + S2z .

(b) Compute the commutators [Sx, Sy], [Sy, Sz] and [Sz, Sx], and express each result through oneof the matrices given above. Hint: [A,B] = AB −BA.

(c) The results from (b) can be compactly summarized in an equation of the form [Si, Sj] = aijkSkfor {i, j, k} ∈ {x, y, z} (with summation over k). Find the tensor aijk.

Example Problem 9: Matrix multiplication [2]Points: (a)[1](M); (b)[1](M)

Let A and B be N ×N matrices with matrix elements aij = Ajδim and bij = Biδ

ij. Remark: Since

the indices i and j are specified on the left, they are not summed over on the right even thoughthe index i appears twice in bij on the right.

(a) For N = 3 and m = 2, write these matrices explicitly in the usual matrix representation andcalculate the matrix product AB explicitly.

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(b) Calculate the product AB for arbitrary N ∈ N and 1 ≤ m ≤ N .

[Total Points for Example Problems: 20]

Homework Problem 1: Gaussian integrals [5]Points: (a)[1](M); (b)[1](M); (c)[1](M); (d)[1](M); (e)[1](M)

(a) Show that the two-dimensional Gaussian integral I =´∞−∞

´∞−∞ dxdy e−(x

2+y2) has the valueI = π. Hint: Use polar coordinates; the radial integral can be solved by substitution.

(b) Now calculate the one-dimensional Gaussian integral I0(a) =´∞−∞ dx e−ax

2(here and below,

a ∈ R, a > 0). Hint: I = [I0(1)]2. Explain why!

Determine the value of the x2n Gaussian integral, In(a) =´∞−∞ dx x2n e−ax

2(with n ∈ N), using

two different methods: (c) repeated partial integration, and (d) repeated differentiation:

(c) Calculate I0, I1 and I2 by using partial integration where required. Then use partial integrationto show that In(a) = 2n−1

2aIn−1(a) holds for all n ≥ 1. Use this relation iteratively to determine

In(a) as a function of a and n.

(d) Show that taking n derivatives of I0(a) with respect to a times yields In(a) = (−1)n dnI0(a)dan

.Then calculate these derivatives for a few small values of n. From the emerging pattern,deduce the general formula for In(a).

(e) Calculate the one-dimensional Gaussian integral with a linear term in the exponent: I0(a, b) =´∞−∞ dx e−ax

2+bx (with a, b ∈ R, a > 0). Hint: Complete the square: Write the exponent in

the form −ax2 + bx = −a(x− C)2 +D and then substitute x = x− C.

Homework Problem 2: Volume calculations with elliptical polar coordinates [2]Points: (a)[1](M); (b)[1](M); (c)[1](A,Bonus)

In the following, use elliptical coordinates in two dimensions, defined as x = µa cosφ, y = µb sinφ,with a, b ∈ R, a > b > 0. Calculate the volume V (a, b, c) of the following objects T , E and C, asa function of the length parameters a, b and c.(a) T is a tent with an elliptical base with semi-axes a and b. The height of

its roof is described by the height function hT (x, y) = c[1− (x/a)2−

(y/b)2].

x

y

z

a b

c

(b) E is an ellipsoid with semi-axes a, b and c, defined by (x/a)2+(y/b)2+(z/c)2 ≤ 1 .

x

y

z

ab

c

(c) C is a cone with height c and an elliptical base with semi-axes a and b. All cross sectionsparallel to the base are elliptical, too. Hint: Augment the elliptical coordinates by anothercoordinate, z (analagously to passing from polar to cylindrical coordinates).

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[Check your answers: if a = 1/π, b = 2, c = 3, then (a) VT = 3, (b) VE = 8, (c) VC = 2.]

Homework Problem 3: Jacobian determinant for spherical coordinates [1]Points: [1](E)

Calculate the Jacobian determinant∣∣∣∂(x1,x2,x3)∂(r,θ,φ)

∣∣∣ for the transformation from Cartesian to spherical

coordinates.

Homework Problem 4: Volume and moment of inertia: cylindrical coordinates [3]Points: (a)[1](M); (b)[2](M); (c)[2](A,Bonus)

Consider the homogeneous rigid bodies C, P and B specified below, each with density ρ0. Use cylin-drical coordinates to compute for each its volume V (a) and moment of inertia I(a) = ρ0

´V

dV d2⊥with respect to the symmetry axis, as functions of the dimensionless, positive scale factor a, thelength parameter R, and the mass of the body, M .

(a) C is a hollow cylinder with inner radius R, outer radius aR, and height 2R. [Check yourresults: VC(2) = 6πR3, IC(2) = 15

6MR2.]

(b) P is a paraboloid with height h = aR and curvature 1/R, defined byP = {r ∈ R3 | 0 ≤ z ≤ h, (x2 + y2)/R ≤ z} [Check your results:VP (2) = 2πR3, IP (2) = 2

3MR2.] x

h

z

(c) B is the bowl obtained by taking a sphere, S = {r ∈ R3| x2 + y2 + (z−aR)2 ≤ a2R2}, with radius aR, centered on the point P : (0, 0, aR)T ,and cutting from it a cone, C = {r ∈ R3| (x2 + y2) ≤ (a− 1)z2, a ≥ 1},symmetric about the z axis, with apex at the origin. [Check your results:VB(43

)= 16

9πR3, IB

(43

)= 14

15MR2. What do you get for a = 1? Why?] x

z

P

Hint: Firtst find, for given z, the radial integration boundaries, ρ1(z) ≤ ρ ≤ ρ2(z), then the zintegration boundaries, 0 ≤ z ≤ zm. What do you find for zm, the maximal value of z?

Homework Problem 5: Volume integral over quarter sphere [2]Points: [2](M)

Use spherical coordinates to calculate the volume integral F (R) =´Q

dV f(r) of the function

f(r) = xy on the quadrant Q, defined by x2 + y2 + z2 ≤ R2 and x, y ≥ 0. Sketch Q. [Check yourresult: F (2) = 64

15.]

Homework Problem 6: Wave functions of the hydrogen atom [4]Points: [2](M); (b)[2](M); (c)[2](M,Bonus)

Show that the volume integral Pnlm =´R3 dV |Ψnlm(r)|2 for the following functions Ψnlm(r) =

Rnl(r)Yml (θ, φ), with spherical coordinates r = r(r, θ, φ), yields Pnlm = 1:

(a) Ψ210(r) = R21(r)Y01 (θ, φ), R21(r) =

re−r/2√24

, Y 01 (θ, φ) =

(34π

)1/2cos θ

(b) Ψ320(r) = R32(r)Y02 (θ, φ), R32(r) =

4 r2e−r/3

81√

30, Y 0

2 (θ, φ) =(

516π

)1/2(3 cos2 θ − 1)

(c) Show that the so-called ‘overlap integral’ O =´R3 dV Ψ320(r)Ψ210(r) yields zero.

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Hint: In =´∞0

dx xn e−x = n! .

Background information: The Ψnlm(r) are quantum mechanical ‘eigenfunc-tions’ of the hydrogen atom, where n, l and m are ’quantum numbers’ whichspecify the quantum state of the system. A particle in this state is found withprobability |Ψnm(r)|2dV within the volume element dV at position r. Thetotal probability for being found anywhere in R3 equals 1, hence Pnlm = 1holds for every eigenfunction Ψnm(r).

The figures each show a surface on which |Ψnlm|2 has a constant value. The eigenfunctions forman orthonormal basis in the space of square-integrable complex functions on R3, hence the volumeintegral of two eigenfunctions vanishes if their quantum numbers are not equal.

Homework Problem 7: Matrix multiplication [2]Points: [2](M)

Compute all possible products of two of the following matrices:

P =

2 0 3

−5 2 7

3 −3 7

2 4 0

, Q =

(−3 1

−1 0

2 1

), R =

(6 −1 4

4 4 −4−4 −4 6

).

Homework Problem 8: Spin 1 matrices [2]Points: (a)[0,5](E); (b)[1,5](E)

The following matrices are used for the description for quantum mechanical particles with spin 1:

Sx = 1√2

(0 1 01 0 10 1 0

), Sy = 1

√2

(0 −i 0i 0 −i0 i 0

), Sz =

(1 0 00 0 00 0 −1

).

(a) Compute S2 = S2x + S2

y + S2z .

(b) Comute the commutators [Sx, Sy], [Sy, Sz] and [Sz, Sx], and express each result through oneof the matrices given above. Hint: [A,B] = AB −BA.

Homework Problem 9: Matrix multiplication [1]Points: (a)[0.5](E); (b)[0.5](E)

Let A and B be N ×N matrices with matrix elements aij = AiδiN+1−j and bij = Biδ

ij. Remark:

Since the indices i and j are specified on the left, they are not summed over on the right eventhough the index i appears twice in bij on the right.

(a) For N = 3 and m = 2, write these matrices explicitly in the usual matrix representation andcalculate the matrix product AB explicitly.

(b) Calculate the product AB for arbitrary N ∈ N and 1 ≤ m ≤ N .

[Total Points for Homework Problems: 22]

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