This document contains 4 physics problems involving quantum mechanics concepts:
1) Calculating the commutator of angular momentum operators and relating the time derivative of angular momentum to torque.
2) Determining spherical harmonic functions using angular momentum operators.
3) Solving for energy levels and eigenfunctions of a rigid rotor system of two particles attached at either end of a massless rod.
4) Verifying commutation relations for spin and angular momentum operators and showing that the cross product of an angular momentum operator with itself is non-zero.
On the Fixed Point Extension Results in the Differential Systems of Ordinary ...BRNSS Publication Hub
In this work, a fixed point x(t), t ∈ [a,b], a ≤ b ≤ +∞ of differential system is said to be extendable to t = b if there exists another fixed point xttaccb()[],,, of the system (1.1) below and xtxttab()=()[),, so that given the system
x’ = f(t,x); f: J × M → Rn
We aim at using the established Peano’s theorem on existence of the fixed point plus Picard–Lindelof theorem on uniqueness of same fixed point to extend the ordinary differential equations whose local existence is ensured by the above in a domain of open connected set producing the result that if D is a domain of R × Rn so that F: D → Rn is continuous and suppose that (t0,x0) is a point D where if the system has a fixed point x(t) defined on a finite interval (a,b) with t ∈(a,b) and x(t0) = x0, then whenever f is bounded and D, the limits
xa
xtta()lim()+=+
xb
xttb()lim()−=
exist as finite vectors and if the point (a, x(a+)),(b, x(b–)) is in D, then the fixed point x(t) is extendable to the point t = a(t = b). Stronger results establishing this fact are in the last section of this work.
On the Fixed Point Extension Results in the Differential Systems of Ordinary ...BRNSS Publication Hub
In this work, a fixed point x(t), t ∈ [a,b], a ≤ b ≤ +∞ of differential system is said to be extendable to t = b if there exists another fixed point xttaccb()[],,, of the system (1.1) below and xtxttab()=()[),, so that given the system
x’ = f(t,x); f: J × M → Rn
We aim at using the established Peano’s theorem on existence of the fixed point plus Picard–Lindelof theorem on uniqueness of same fixed point to extend the ordinary differential equations whose local existence is ensured by the above in a domain of open connected set producing the result that if D is a domain of R × Rn so that F: D → Rn is continuous and suppose that (t0,x0) is a point D where if the system has a fixed point x(t) defined on a finite interval (a,b) with t ∈(a,b) and x(t0) = x0, then whenever f is bounded and D, the limits
xa
xtta()lim()+=+
xb
xttb()lim()−=
exist as finite vectors and if the point (a, x(a+)),(b, x(b–)) is in D, then the fixed point x(t) is extendable to the point t = a(t = b). Stronger results establishing this fact are in the last section of this work.
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MATLAB sessions: Laboratory 6
MAT 275 Laboratory 6
Forced Equations and Resonance
In this laboratory we take a deeper look at second-order nonhomogeneous equations. We will concentrate
on equations with a periodic harmonic forcing term. This will lead to a study of the phenomenon known
as resonance. The equation we consider has the form
d2y
dt2
+ c
dy
dt
+ ω20y = cosωt. (L6.1)
This equation models the movement of a mass-spring system similar to the one described in Laboratory
5. The forcing term on the right-hand side of (L6.1) models a vibration, with amplitude 1 and frequency
ω (in radians per second = 12π rotation per second =
60
2π rotations per minute, or RPM) of the plate
holding the mass-spring system. All physical constants are assumed to be positive.
Let ω1 =
√
ω20 − c2/4. When c < 2ω0 the general solution of (L6.1) is
y(t) = e−
1
2 ct(c1 cos(ω1t) + c2 sin(ω1t)) + C cos (ωt− α) (L6.2)
with
C =
1√
(ω20 − ω2)
2
+ c2ω2
, (L6.3)
α =
⎧
⎨
⎩
arctan
(
cω
ω20−ω2
)
if ω0 > ω
π + arctan
(
cω
ω20−ω2
)
if ω0 < ω
(L6.4)
and c1 and c2 determined by the initial conditions. The first term in (L6.2) represents the complementary
solution, that is, the general solution to the homogeneous equation (independent of ω), while the second
term represents a particular solution of the full ODE.
Note that when c > 0 the first term vanishes for large t due to the decreasing exponential factor.
The solution then settles into a (forced) oscillation with amplitude C given by (L6.3). The objectives of
this laboratory are then to understand
1. the effect of the forcing term on the behavior of the solution for different values of ω, in particular
on the amplitude of the solution.
2. the phenomena of resonance and beats in the absence of friction.
The Amplitude of Forced Oscillations
We assume here that ω0 = 2 and c = 1 are fixed. Initial conditions are set to 0. For each value of ω, the
amplitude C can be obtained numerically by taking half the difference between the highs and the lows
of the solution computed with a MATLAB ODE solver after a sufficiently large time, as follows: (note
that in the M-file below we set ω = 1.4).
1 function LAB06ex1
2 omega0 = 2; c = 1; omega = 1.4;
3 param = [omega0,c,omega];
4 t0 = 0; y0 = 0; v0 = 0; Y0 = [y0;v0]; tf = 50;
5 options = odeset(’AbsTol’,1e-10,’RelTol’,1e-10);
6 [t,Y] = ode45(@f,[t0,tf],Y0,options,param);
7 y = Y(:,1); v = Y(:,2);
8 figure(1)
9 plot(t,y,’b-’); ylabel(’y’); grid on;
c⃝2011 Stefania Tracogna, SoMSS, ASU 1
MATLAB sessions: Laboratory 6
10 t1 = 25; i = find(t>t1);
11 C = (max(Y(i,1))-min(Y(i,1)))/2;
12 disp([’computed amplitude of forced oscillation = ’ num2str(C)]);
13 Ctheory = 1/sqrt((omega0^2-omega^2)^2+(c*omega)^2);
14 disp([’theoretical amplitude = ’ num2str(Ctheory)]);
15 %----------------------------------------------------------------
16 function dYdt = f(t,Y,param)
17 y = Y(1); v = Y(2);
18 omega0 = param(1); c = param(2); omega = param(3);
19 dYdt = [ v ; cos(omega ...
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My paper for Domain Decomposition Conference in Strobl, Austria, 2005Alexander Litvinenko
We did a first step in solving, so-called, skin problem. We developed an efficient H-matrix preconditioner to solve diffusion problem with jumping coefficients
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2. Problem # 15.1 : Torque
(a) Calculate the commutator [Lx , p2 ].
(b) Prove that for a particle in a potential V (r) the rate of change of the expectation value
of the orbital angular momentum L is equal to the expectation value of the torque:
d
L = N ,
dt
where
N = r × (− V ) .
This is the rotational analog to Ehrenfest’s theorem.
(c) Show that d L /dt = 0 for any spherically symmetric potential. This is one way of
proving the conservation of angular momentum.
3. Problem # 15.2 : Spherical harmonics
(a) What is L+ Yll ? (No calculation allowed!)
(b) Use the result of (a), together with
∂ ∂
L+ = eiφ + icos(θ) and Lz Yll = lYll ,
∂θ ∂φ
to determine Yll (θ, φ) up to a normalization constant.
(c) Determine the normalization constant by direct integration.
4. Problem # 15.3 : Rigid rotor
Two particles of mass m are attached to the ends of a massless rigid rod of length a. The
system is free to rotate in three dimensions about the center of mass. Assume the center of
mass is at rest.
(a) Show that the allowed energies of this rigid rotor are
2
l(l + 1)
El = , l = 0, 1, 2, . . .
ma2
Hint: First express the classical energy in terms of the total angular momentum.
(b) What are the normalized eigenfunctions for this system? What is the degeneracy of the
nth energy level?
5. Problem # 15.4 :
(a) Verify that the components of the spin-1/2 matrices defined as S = ( /2)σ, where σ
is the vector of Pauli matrices, satisfy the commutation relations
[Sx , Sy ] = i Sz , [Sy , Sz ] = i Sx , [Sz , Sx ] = i Sy .
(b) Verify that the components of the L = 1 matrices defined as
0 1 0 0 −i 0 1 0 0
Lx = √ 1 0 1 , Ly = √ i 0 −i , Lz = 0 0 0 ,
2 0 1 0 2 0 i 0 0 0 −1
satisfy the commutation relations
[Lx , Ly ] = i Lz , [Ly , Lz ] = i Lx , [Lz , Lx ] = i Ly .
(c) In ordinary vector algebra, the cross product of a vector with itself is always zero:
A × A = 0. When the vectors are operators, however, this can be different. Show that
the angular momentum operator behaves as L × L = i L.
![Problem # 15.1 : Torque
(a) Calculate the commutator [Lx , p2 ].
(b) Prove that for a particle in a potential V (r) the rate of change of the expectation value
of the orbital angular momentum L is equal to the expectation value of the torque:
d
L = N ,
dt
where
N = r × (− V ) .
This is the rotational analog to Ehrenfest’s theorem.
(c) Show that d L /dt = 0 for any spherically symmetric potential. This is one way of
proving the conservation of angular momentum.](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)