Mathematical Methods of Physics – Fall 2010 – Dr. E.J. Zita
Week 8 – Thursday 18 Nov. 2010
We missed Tuesday due to the power outage, so today we’ll focus on
Modern Physics (Schrodinger Equation – Ch.5) and Mechanics (Central Forces – Ch.6)
First: outstanding Questions from last week?
Mechanics Ch 4: Oscillations PIQs
Points: Automatic Unicorns
For a nonisotropic oscillator, if the angular frequencies are commensurate ( w1/n1 =w2/n2=
w3/n3) the path of the oscillator is closed and forms a Lissajous figure. Otherwise it is not
closed and it fills the entire rectangular box 2A, 2B, 2C, where A, B and C are the amplitudes
of the oscillators three components. (page 171) This is awesome!
Analogously to two dimensional motion, the work done in moving a particle from one place
to another in three dimensions is equal to the difference between the particle's final kinetic
energy and initial kinetic energy. (page 146)
Insights:
Constrained motion of a particle is similar to the constrained motion of a pendulum bob. For
a particle constrained on a frictionless surface the force only acts tangent to the surface,
similarly, for a pendulum bob the force only acts perpendicular to the line of the string, that
is, tangent to the arc that the pendulum bob traces in the air.
We did not know that in a world with friction, projectiles travel the furthest when launched at
an angle less than 45 degrees. Our calculus textbooks were written in frictionless worlds.
Lissajous figures do not have to be plain looking:
Questions:
On page 150, Fowles and Cassiday write, "a vanishing curl F ensures that the line integral of
F is zero and, thus, that F is a conservative force" but on page 156, in the last paragraph after
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talking about forces with curl zero, they write that separable forces that have time dependent
components aren’t necessarily conservative. Are they still talking about forces with curl
zero?
On page 172, in example 4.4.1, why do we write x= Acos( ω t)+Bsin( ω t) instead of
x=Acos( ω t)?
How are the paths of pendulums different from cycloids?
The Other 7%
Q2. What does transposing the square term mean on pg 169? Mechanics
3. How do you solve the equations on pg. 159? What are they doing there?
P1.A general quadratic equation represents an ellipse, a parabola or a hyperbola
if its discriminant (b^2 - 4ac) is negative (ellipse), zero ( parabola) or positive ( hyperbola).
2. If forces are separable you can do the diff EQ in the separate co-ordinate directions and
link them by time to find the max distance or optimal angle.
1. In a non isotropic operator if you don't have n1 n2 and n3 as integers you will fill a box
equal to the mag of oscillation in all three directions and centered around the origin.
2. In a path dependent function there is no closed loop where energy is conserved.
3. Because the force acting on a particle in a magnetic field is equal to the cross product of its
velocity and the magnetic field it is in multiplied by the charge of the particle it must move in
a direction perpendicular to the magnetic field because the cross product introduces a third
dimension z perpendicular to the x and y planes.
Team Physics: P1: Because the linear resistive force separates into components along each
axis, the acceleration can also be separated into equations for each axis.(Mechanics 161)
P2: Properties of projects motion: (1)max height of projectile (2)the time it takes to reach
maximum height (3)time of flight of projectile (4)range and max range(Mechanics 160)
P3: The Stokes' Theorem states that the closed loop line integral of any vector F*n da
integrated over a surface S.(Mechanics 150)
I1:The two forces along the projectile act along the vertical direction(gravity) and along its
instantaneous velocity.(Mechanics 159)
I2: The total energy does not remain constant but rather depends on the nonconservative
force. (Mechanics 153)
I3:Isotropic oscillator is when the restoring force is independent of the direction of
displacement while a nonisotropic oscillator has the magnitude of the restoring force depend
on the direction of the displacement. (170)
Q2: How does the mass of the object flying affect the conservation of total energy? Does it
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diminish faster if the object is lighter or heavier?
Q3: What is the gradient of a function? (151)
IPL:
Q3: On page 153 of the mechanics books it states "The total energy E does not remain
constant throughout the motion of the particle but increases and decreases..." How does the
total energy of a particle increase with no extra forces applied to it?
P1: Numerical approximations of the Lorenz equations are extremely sensitive to your initial
conditions.
P2: For cases with a 'smooth constraint' the 'R' portion of your force equation vanishes after
the dot product.
I1: Aside from representing the “curl” of the vector field, Del can also denote the divergence
of a scalar or vector field depending on how it is used.
I3: For a three dimensional Isotropic Harmonic Oscillator the six constraints of integration
are determined by the initial conditions of the system.
RR: Points:
1.) A matrix with a zero determinant is called degenerate. A degenerate matrix is not
invertible.
2.)Eigenvalues are the roots of the characteristic polynomial of a system. These root values
can be real or complex.
3.) A spiral sink is when solutions spiral inwards toward the origin, a spiral source is when
solutions flow out from the origin in a spiral.
Insights:
1.)Resonance is achieved when a system is able to store and quickly transfer energy between
it's storage modes. This cause the system to have a greater amplitude at certain frequencies.
These frequencies are the resonance frequencies.
2.)When a zero Eigenvalue exists in a system, there will be a Line of equilibrium points. All
solutions should tend toward this line.
3.) The period of a system that exhibits simple harmonic motion, is independent of initial
conditions.
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Modern Physics - Quantum mechanics: Ch.5: Schrodinger Eqn

Discuss assumptions of SE – derive time-independent form

Derive the rest of PIB (did half of it last week)

Matching BC

Physlet demos of wave scattering, particle in box, tunneling – PREDICT FIRST

Peer instruction – groups read and discuss examples p.164 – then describe to class.

DO SHO

Discuss questions p.168

Do prob.2, 4, 8, choose HW (NB on 11 – need to look up integral)
Discuss PIQs:
P1: To normalize the probability function set the definite integral from negative infinity to
infinity of the probability density function P(x)dx=|ψ(x)|2dx equal to one and solve for A.
Normalizing guarantees a 100% probability that the particle will be somewhere within the
range. The probability that a particle will be found between two points x1 and x2 is given by
the definite integral of the normalized probability function. (p145) (AU)
P2: The solution of the Schrödinger equation for a particle trapped in a box will be a series of
standing waves, like a string fixed at both ends. This means that only certain energy states
can occur. The lowest energy state is called the ground state, higher states are called excited
states. (p148-149) (AU)
p1. Schrodinger's equation must follow the criteria that it conserves energy. It is a linear,
single valued equation and it is consistent with the de Broglie's hypothesis. (Mdrn, p.139)
(SS)
p2. A normalization condition shows us how to find the constants in a wave function. (Mdrn,
p. 145) (SS)
p3. Only a properly normalized wave function can be used for a physical meaningful
calculation. (Mdrn, p. 145) (SS)
P - Recreating systems with identical properties does not ensure that the system will behave
the same way from one experiment or measurement to the next. Classical mechanics would
have us believe otherwise. (Krane p.126) (Underwater)
- Krane notes that "any system in a potential energy minimum behaves approximately like a
simple harmonic oscillator." Because of this, we can use Schrodinger's equation to handle the
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situation of a SHO, although we will never find an example of a quantum oscillator in nature.
(155) (Underwater)
I1: Krane writes, “When we speak of a position, we are referring to a particle; when we
speak of motion from L/4 to 3L/4, we are dealing with waves.” (p149) This is like
performing the double slit experiment with photographic film as the screen – the light acts as
a wave when it is traveling from the source through the slits to the photographic film, but
when it hits the photographic film it’s position is suddenly defined and it acts like a particle
by changing oxidation state of one molecule of the film. (AU)
I2: Normalization requires the use of boundary conditions. The definite integral from
negative infinity to infinity of the square of the absolute value of Asin(x) is infinity, which
means that regardless of the value of A, the definite integral will never be one unless the
particle is forbidden from some portions of the x axis. (AU)
I3: On page 154 Krane writes that “occasionally it happens that two different sets of quantum
numbers nx and ny have exactly the same energy. This situation is known as degeneracy.”
This connects the ideas of quantum energy states to the eigenvalue degeneracy that we see in
3.11 of Boas. We would like to learn more about this. (AU)
I. It is impossible to make simultaneous measurements of a particles location and momentum
or a particles energy and time coordinate (Mdrn, p. 115) (SS)
P - Recreating systems with identical properties does not ensure that the system will behave
the same way from one experiment or measurement to the next. Classical mechanics would
have us believe otherwise. (Krane p.126) (Underwater)
- Krane notes that "any system in a potential energy minimum behaves approximately like a
simple harmonic oscillator." Because of this, we can use Schrodinger's equation to handle the
situation of a SHO, although we will never find an example of a quantum oscillator in nature.
(155) (Underwater)
Q1: On page 152 Krane writes that “inside the box we consider solutions that are separable.”
Is this for convenience or is there something about a particle in a two dimensional box that
leads to separable equations? (AU)
Q2: In the beginning of chapter 5 Krane writes that in deriving Schrödinger’s equation we
use classical mechanics (139). Is there a relativistic Schrödinger equation for situations
where classical mechanics doesn’t make sense? (AU)
q1. Schrödinger's equation gives the probability for finding a wave at a given location, can
this be applied to macroscopic waves? (Mdrn) (SS)
q2. What does the lowercase k symbol mean. (Mdrn. 113). (SS)
Q What is a quantum number exactly? (Underwater)
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Boas: Discuss PIQs:
P: The total differential dz is equal to ∂z/∂x*dx+∂z/∂y*dy. If z =f(x,y) describes a surface
then dz describes the change in the surface due to changes dx and dy in x and y. If the partial
derivatives ∂z/∂x and ∂z/∂y are continuous and dx and dy are are small then dz is a good
approximation to ∆z. (AU)
I1. It is possible to expand multi-variable functions into power series. The general form of
which looks very similar to foiling binomials. (Boas, p. 192) (SS)
q3. Equation 3.6 on p. 194 of Boas, can we get clarification on the geometrical meaning of
the total differential of z? (SS)
Q: On page 193 Boas writes that dx =∆x but that "dy is the tangent approximation (or linear
approximation) to ∆y." What does this mean? Why do we need to differentiate (no pun
intended) between dx and dy in this way for our definitions to work? (AU)
Q - What does it mean for the differential dz to be called the total differential of z? (Boas p.
194) (Underwater)
New DiffEq PIQs:
(DiffEq, pg 341) The graph is a good tool to start recognizing trends in phase portraits (UW)
Questions
Q1 - What real life applications does solving linear systems in three dimensions is a
necessity? (354) (TP)
Q2: Can we go over how eigenvalues are used in real applications, both real and imagined?
(TP)
Insights
(pg 361, DiffEq) Notice how phase space is similar to magnetic field moving with respect to
current moving through a wire. (UW)
I: Even though the example on 363 contains many zero entries, the system does not easily
decouple into lower dimensions (363 ). (TP)
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Mechanics – Central Forces Ch.6
Galileo (1564-1642), Newton (1642-1727)
Kepler (1571-1630)

Newton’s laws generalized Galileo’s; we can derive Kepler’s laws from Newton’s.

G and N calculated nearly the same g (inclined plane / moon’s orbit)

N: inertial mass = gravitational mass!

F = GmM/R2 – review spherical coordinates.

Newton’s shell theorem p.223-225

Kepler’s laws – derive K3 from N2

HW p.259 # 3, 4, 6, 7

Conic sections p.235

p.243-244: planets follow Keplerian velocity curve (6.6.11). Stars in galaxies do
NOT. Therefore the mass in galaxies is not centrally concentrated, as it is in a solar
system.

HW # 9, 16, and find the mass distribution (r) required for a galactic velocity
distribution to flatten out (except near r=0).

Section 6.14: Scattering of alpha particles p.264
New Mechanics PIQs:
P1 - Angular momentum is rotational momentum of a particle about an origin. It is
perpendicular to the plane of rotation and defined as L = r x p, where r is the radius of the
orbit and p is the particle's linear momentum tangential to the path of the orbit. Since the
cross product defines a plane orbits can be considered two dimensionally. (p226-227) (AU)
P2 - A central force is a force whose "lines of action either emanate from or terminate on a
single point or center." An isotropic force is a force whose magnitude is the same in all
directions. (p220) (AU)
P3- Figure 6.11.2 shows how the relationship between total energy and the effective potential
energy shapes the orbits of bodies interacting under gravity. (p259) The total energy of a
particle determines the semimajor axis of its orbit. If E<0 we see closed orbits, if E=0 we get
a parabolic orbit and if E>0 we see a hyperbolic orbit. (p253) (AU)
P1 (Mech, p.174) - The differential equation of motion of a particle moving in a purely magnetic
field is given by equation 4.5.6. It states that the acceleration of the particle is always at right
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angles to the direction of motion. This means that the tangential component of the acceleration is
zero, and so the particle moves with constant speed. This is true even if the magnetic field is a
varying function of the position r, as long as it does not vary with time. (UW)
P2 (Mech) A circular orbit acts as a harmonic oscillator. if disturbances are weaker than
restoring force than a stable orbit is maintained (UW)
P1 - Kepler’s three laws of planetary motion are 1) Law of Ellipses, 2) Law of Equal Areas,
and 3) The Harmonic Law . (O7%)
P2 - For a degenerate system, the solution will have a general orm of: Y(t) = e^(gam*t)V0 +
te^(gam*t)V1 and V0 and V1 are related by: V1 = (A-gam*I)V0. (diff eq p.313) (O7%)
P3 - Because the earth is rotating its equator radius is 13 miles larger than its polar radius.
(O7%)
P1 - Uniformly dense spherical objects attract other objects as if all mass is concentrated at
the center of mass. (RR)
P2 - Observable Planetary motion does not fit Kepler’s model due to the non ideal nature or
the system ( the solar system does not have uniformly dense spheres and each planet’s mass
effects the other planet’s motion) (RR)
P3 - Kepler derived his three laws from Newton’s universal gravitation and mechanics. (RR)
P1: "Every object in the universe must attract (albeit very weakly, in most cases) every other
object in the universe" (233 Mechanics). (TP)
P2: For linear systems with two dependent variables, the behavior of solutions and the nature
of the phase plane can be determined by computing the eigenvalues and eigenvectors of the
2x2 coefficient matrix (254 Mechanics). (TP)
P3: Kepler's three laws of planetary motion are (1) law of ellipses (each planet travels in an
ellipse) (2) law of equal areas (a line drawn between the sun and planet sweeps on equal ares
in equal times) and (3) harmonic law (the square of the sidereal period of a planet is
proportional to the cube of its semimajor axis) (Mechanics 225). (TP)
Insights
I1 - Through investigating the apparent failures of Keplers laws to fully explain the orbit of
Mercury, Einstein pre-verified general relativity. (264) The authors claim that this was the
case of a "highly remarkable event of a discrepancy between observation and existing theory
leading to the confirmation of an entirely new superceding theory." This doesn't seem
remarkable to us. Through our seminar reading we have learned of other cases where this
happened -- for example in the Michelson Morley experiments. (AU)
I2 - On page 265, the authors write that "the domain of Newtonian physics would seem to be
merely limited, rather than 'wrong,' and its practitioners from that time on would now have to
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be aware of those limits." This reflects Lederman's assertion in the God Particle that the
discovery of new theories doesn't necessitate discarding the old theories, rather, the new
theories encompass the old ones. (AU)
I3 - The fact that spiral galaxies don't seem to follow Kepler's laws led to the theories of dark
matter. (p244) If we find direct evidence of dark matter, it would be another case in which
examining the way in which a theory fails leads to new discoveries, perhaps we will even
gain a new theory about matter! (AU)
I1 - (Mech) The momentum of the earth as it moves away from the sun (early summer and
end of winter) is a damping force; the gravity of the sun is the restoring force for our orbit.
(UW)
I2 (Mech, p.179) - Periodic motion is said to be isochronous when the particle undergoes
periodic motion whose frequency is independent of the amplitude of oscillation, unlike the
simple pendulum for which the frequency depends on the amplitude. (UW)
I1 - Because of the inertial reaction -mA(naught) the plumb line goes towards the center of
the earth. The plumb line is always perpendicular to the surface of the earth. (O7%)
I2 - For differential equations that do not exhibit straight line solutions we need to use
complex algebra to find the solutions. However the algebraic techniques we learned before
are still useful, we just need to use Euler's formula to find real solutions. (O7%)
I3 - An example of the elegance of the mathematic order of the universe is Kepler’s
Harmonic Law. (O7%)
I1: The diagonal matrix is similar to the identity matrix, where besides the diagonal, each
column and row contains 0. (258 Mechanics). (TP)
I3: Kepler believed that the world was a fundamentally beautiful place, which led him to
believe that there was a true harmony to planetary motion (Mechanics 238). (TP)
Questions:
Q1 - What would be an example of a force that is central but not isotropic? (AU)
Q1 - Can we go over the derivation of Kepler’s equations? (RR)
Q2 - What real world observation can be made to detect a system that would have complex
eigenvalues? (RR)
Q2 - On page 236 the authors write that, in the case of the nearly circular orbits of most of
the large bodies in our solar system, "it is difficult to imagine how any natural process could
have established such nearly perfect prerequisites." Is it the case that bodies with more
elliptical orbits were more likely to suffer catastrophic collisions or experience gravitational
interactions that flung them out of the solar system? (AU)
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Q3 - Can we read chapter 5 next? (A friend of mine who doesn't want his or her name
mentioned accidentally read chapter five and wants to know) (AU)
Q1 - If gravity attracts objects to the centers only , then what would happen if I went to the
center of the earth? Would I feel a gravitational force pulling from the mass on the surface (I
realize that this value would be much less than gravity pull of the center)? (UW)
Q2 - Why does the equation for a simple elliptical orbit look like the equation for a simple
harmonic oscillator? What part of the equation is accounting for the change in arc? (UW)
Q3 - Why is there not a type of linear system called a "spiral saddle"? What's the difference
between centers and saddles? (UW)
Q4 - What kind of systems do functions with complex Eigenvalues describe? (UW)
Q1 - Why is it that for a system with real values and complex eigenvalues, the eigenvalues
are related such that you only have to compute the general solution with one of them instead
of both? (diff eq. p 296) (O7%)
Q2 - Can we go over the different types of acceleration in class on pg 193? (O7%)
Q3 - Are there any phenomena, like orbital patterns of planets, that don’t fit with
mathematics? (No recognizable pattern to describe mathematically?) Is this then called
chaotic? (O7%)
Q: Why in equation 6.2.8 the right hand term has a negative? (225 Mechanics) (TP)
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