Roles of Planck, Bohr, Heisenberg, and Schrödinger in Physics - Prof. Donald G. Luttermose, Study notes of Physics

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Physics 2018: Great Ideas in Science:

The Physics Module

Quantum Mechanics Lecture Notes

Dr. Donald G. Luttermoser East Tennessee State University

Edition 1.

Abstract

These class notes are designed for use of the instructor and students of the course Physics 2018: Great Ideas in Science. This edition was last modified for the Fall 2007 semester.

iii) In order to explain the periodic “backward” (i.e., retrograde) motion of the planets on the sky, Claudius Ptolemy, who lived around 140 A.D. and a firm believer in Aristotle’s philosophy, developed a geo- centric system that had the planets revolving on smaller circles (called epicycles) whose “centers” orbited the Earth (with the larger circular orbits called deferents ).

iv) In 1543, Nicholas Copernicus (1473–1543), a Pol- ish astronomer and cleric, published his heliocen- tric model for the solar system where Earth was a planet, similar to the other planets, in circular orbit about the Sun =⇒ the Copernican Revo- lution.

v) Johannes Kepler (1571–1630), a German mathe- matician and astronomer, modified the Copernican model by having the planets orbit the Sun in el- liptical and not circular paths when he formulated the three laws of planetary motion.

c) Invention of the scientific method owes much to the work of Galileo Galilei (1564–1642), an Italian astronomer and physicist. Galileo is considered to be the father of experimental physics. i) Determined that objects of different masses fall at the same rate on the Earth’s surface (which con- tradicted the teachings of Aristotle).

ii) Came up with the concept of the pendulum clock.

iii) Developed the various concepts of motion.

iv) First to use the telescope to study the cosmos =⇒ discovered the 4 large moons of Jupiter (i.e., the Galilean moons), that Venus goes through phases (like our Moon), that the Moon’s surface wasn’t smooth, and that dark spots appear on the Sun (i.e., sunspots) from time to time.

d) Isaac Newton (1642–1727), an English astronomer and physicist, was perhaps the greatest scientist whoever lived! The work he did is often referred to as the Newtonian Revolution. i) Invented calculus to describe his physics.

ii) Developed the laws of motion.

iii) Developed the law of gravity.

iv) Invented the reflecting telescope.

v) Developed many theories in optics and showed that white light is composed of the rainbow of col- ors.

e) James Clerk Maxwell (1831–1879) was a Scottish mathe- matician and theoretical physicist from Edinburgh, Scot- land and had two major impacts on physics. i) His most significant achievement was developing a set of equations that showed how electricity and magnetism are related =⇒ Maxwell’s equations. These equations merged the electric force and the

observable quantities, for example the position and momentum of a particle, has an unavoidable un- certainty. Together with Bohr, he formulated the Copenhagen interpretation of quantum mechanics.

iv) In 1926 German physicist Erwin Schr¨odinger (1887–1961) published a paper on wave mechanics and what is now known as the Schr¨odinger equa- tion. In this paper he gave a “derivation” of the wave equation for time independent systems, and showed that it gave the correct energy eigenvalues for the hydrogen-like atom.

v) There are others that we could cite here, but the above four are the most important.

i) From 1970 through 1973, particle physicists developed the Standard Model of particle physics which describes three of the four known fundamental interactions between the elementary particles that make up all matter. i) A large number of physicists were responsible for its development.

ii) To date, almost all experimental tests of the three forces described by the Standard Model have agreed with its predictions.

iii) Through the Standard Model all of the large va- riety of so-called “elementary” particles that have been discovered in particle accelerators can be ex- plained as a composite of any of six quarks and six leptons.

B. The Nature of Physics.

1. 2 main branches: a) Classical Physics: i) Classical Mechanics (also called Newtonian Mechanics).

ii) Thermodynamics (the study of heat).

iii) Fluid Mechanics (the study of fluids).

iv) Electromagnetism (the study of electricity and magnetism).

v) Optics (the interaction of light with lenses and mirrors).

vi) Wave Mechanics (the study of wave motion).

b) Modern Physics: i) Special Relativity and General Relativity.

ii) Quantum Mechanics (also called Atomic Physics).

iii) Nuclear Physics.

iv) Statistical Mechanics (thermodynamics in terms of probabilities).

v) Condensed Matter (once called Solid State Physics).

d) Models: A representation of a physical system (e.g., the Bohr model atom).

e) Hypothesis: The tentative stages of a model that has not been confirmed through experiment and/or observa- tion (e.g., Ptolomy’s model solar system).

f) Theory: Hypotheses that are confirmed through repeated experiment and/or observation (e.g., Newton’s theory of gravity). The word “theory” has different meanings in common English (i.e., it can mean that one is making a guess at something). However, it has a very precise meaning in science! Something does not become a theory in science unless it has been validated through repeated experiment as described by the scientific method.

4. At this point, we will differences between the classical view of physics and the quantum view of physics.

C. The Classical Point of View.

1. A system is a collection of particles that interact among them- selves via internal forces and that may interact with the world outside via external fields. a) To a classical physicist, a particle is an indivisible mass point possessing a variety of physical properties that can be measured. i) Intrinsic Properties: These don’t depend on the particle’s location, don’t evolve with time, and aren’t influenced by its physical environment (e.g., rest mass and charge).

ii) Extrinsic Properties: These evolve with time in response to the forces on the particle (e.g., posi- tion and momentum).

b) These measurable quantities are called observables.

c) Listing values of the observables of a particle at any time =⇒ specify its state. (A trajectory is an equivalent way to specify a particle’s state.)

d) The state of the system is just the collection of the states of the particles comprising it.

2. According to classical physics, all properties, intrinsic and ex- trinsic, of a particle could be known to infinite precision =⇒ for instance, we could measure the precise value of both position and momentum of a particle at the same time.
3. Classical physics predicts the outcome of a measurement by cal- culating the trajectory (i.e., the values of its position and mo- mentum for all times after some initial (arbitrary) time t◦) of a particle: {~r(t), ~p(t); t ≥ t◦} ≡ trajectory, (I-1) where the linear momentum is, by definition,

~p(t) ≡ m d dt ~r(t) = m ~v(t) , (I-2)

with m the mass of the particle. a) Trajectories are state descriptors of Newtonian physics.

b) To study the evolution of the state represented by the trajectory in Eq. (I-1), we use Newton’s Second Law: ∑ (^) ~ F = m~a , (I-3)

D. The Quantum Point of View.

1. The concept of a particle doesn’t exist in the quantum world — so-called particles behave both as a particle and a wave =⇒ wave-particle duality. a) The properties of quantum particles are not, in general, well-defined until they are measured.

b) Unlike the classical state, the quantum state is a conglom- eration of several possible outcomes of measurements of physical properties.

c) Quantum physics can tell you only the probability that you will obtain one or another property.

d) An observer cannot observe a microscopic system without altering some of its properties =⇒ the interaction is un- avoidable : The effect of the observer cannot be reduced to zero, in principle or in practice.

2. This is not just a matter of experimental uncertainties, nature itself will not allow position and momentum to be resolved to infinite precision (see Figure I-1) =⇒ Heisenberg Uncertainty Principle (HUP):

∆x(t◦) ∆px(t◦) ≥

h 2 π

¯h 2

, (I-6)

where h = 6. 62620 × 10 −^27 erg-sec = 6. 626 × 10 −^34 J-sec is Planck’s Constant. a) ∆x(t◦) is the minimum uncertainty in the measurement of the position in the x-direction at time t◦.

b) ∆px(t◦) is the minimum uncertainty in the measurement of the momentum in the x-direction at time t◦.

∆x

x

∆p

p

Figure I–1: The results of measurement of the x components of the position and momentum of a large number of identical quantum particles. Each plot shows the number of experiments that yield the values on the abscissa. Results for each component are seen to fluctuate about a central peak, the mean value 〈x〉 and 〈p〉.

c) Similar constraints apply to the pairs of uncertainties ∆y(t◦), ∆py (t◦) and ∆z(t◦), ∆pz (t◦).

d) Position and momentum are fundamentally incompatible observables =⇒ the Universe is inherently uncertain!

e) The HUP strikes at the very heart of classical physics: the trajectory =⇒ obviously, if we cannot know the position and momentum of a particle at t◦, we cannot specify the initial conditions of the particle and hence cannot calcu- late the trajectory.

f) Once we throw out trajectories, we can no longer use New- ton’s Laws, new physics must be invented!

3. Since Newtonian and Maxwellian physics describe the macro- scopic world so well, physicists developing quantum mechanics demanded that when applied to macroscopic systems, the new physics must reduce to the old physics =⇒ this Correspon- dence Principle was coined by Niels Bohr.
4. Due to quantum mechanics probabilistic nature, only statisti- cal information about aggregates of identical systems can be ob- tained. Quantum mechanics can tell us nothing about the behav- ior of individual systems. Moreover, the statistical information provided by quantum theory is limited to the results of measure- ments =⇒ thou shall not make any statements that can never be verified.

E. Blackbody Radiation

1. In the early part of the 20th century, Max Planck asked the ques- tion: What is the spectrum of electromagnetic (EM) radiation in- side a heated cavity? More specifically, how does this spectrum depend on the temperature T of the cavity, on its shape, size, and chemical makeup, and on the frequency ν of the EM radiation in it? a) Earlier in the mid-19th century, Kirchhoff found that the energy inside such a cavity is independent of the physical characteristics of the cavity (i.e., size and shape), only ν and T were important.

b) Planck was interested in the energy density in the cavity and sought an expression for the radiative energy den- sity per unit volume ρ(ν, T ) and this density in the frequency range ν to ν + dν: ρ(ν, T ) dν.

c) Kirchhoff called his model of a heated cavity in thermal equilibrium a “black-body radiator.” A blackbody is

simply anything that absorbs all radiation incident upon it. Thus a blackbody radiator neither reflects nor trans- mits energy; it just absorbs or emits it.

2. Wien had already experimentally ascertained that the radiative energy density of a blackbody was proportional to ν^3 and, from the work of Stefan, that the integrated energy density ∫ (^) ∞ 0 ρ(ν, T^ )^ dν is proportional to T 4. a) Planck realized that ρ(ν, T ) could not solely depend upon ν^3 since this would imply that the energy density would blow up at small frequencies (i.e., long wavelengths).

b) Planck focused on the exchange of energy between the radiation field and the walls of the cavity. i) He developed a simple model of this process by imagining that the molecules of the cavity walls are resonators — electrical charges undergoing simple harmonic motion.

ii) As a consequence of their oscillations, these charges emit EM radiation at their oscillation frequency, which at thermal equilibrium, equals the frequency ν of the radiation field.

iii) According to classical electromagnetic theory, en- ergy exchange between the resonators and the en- ergy field is a continuous process =⇒ the oscillators can exchange any amount of energy with the field, provided that the energy is conserved in the pro- cess.

d) Having made this assumption, Planck easily derived the radiation law:

ρ(ν, T ) = 8 πν^2 c^3

hν ehν/kT^ − 1

, (I-9)

where k is the above mentioned Boltzmann’s constant. As can be seen, Eq. (I-9) agrees with the empirical relation expressed in Eq. (I-8).

e) The radiative energy density, ρ(ν, T ), is related to the monochromatic radiative energy flux Bν (T ) (i.e., the “brightness” of a glowing object) with the relation

ρ(ν, T ) = 4 π c Bν^ (T^ )^.^ (I-10) f) As such, the monochromatic energy flux (or brightness) of a blackbody is

Bν(T ) = 2 hν

(^3) /c 2 ehν/kT^ − 1

(I-11)

in frequency space, where Bν is measured in J/s/m^2 /Hz/sr (‘sr’ is the steradian unit) in SI units and erg/s/cm^2 /Hz/sr in the cgs unit system. Since Bν dν = Bλ dλ and ν = c/λ, we can also write this function in wavelength space as

Bλ(T ) = 2 hc^2 /λ^5 ehc/λkT^ − 1

. (I-12)

Both Eqs. (I-11) and (I-12) are called the Planck func- tion (in frequency and wavelength space, respectively).

3. Planck’s radiation law not only solve the problem of blackbody radiation, it also opened the door to a new understanding of radi- ation energy in physics =⇒ quantum physics, also called quantum mechanics.

F. The Semi-Empirical Model of Hydrogen.

1. Work that lead to an understanding of the spectrum of the hy- drogen atom took place at the end of the 19th and beginning of the 20th century. As such, much of what of the work described in this and the next few subsections is presented in the cgs unit system since those are the units that were being used in physics at the time.
2. Rydberg (1890), Ritz (1908), Planck (1910), and Bohr (1913) were all responsible for developing the theory of the spectrum of the H atom. A transition from an upper level m to a lower level n will radiate a photon at frequency νmn = c RA Z^2

n^2 −^

m^2

) , (I-13) where the velocity of light, c = 2. 997925 × 1010 cm/s, Z is the effective charge of the nucleus (ZH = 1, ZHe = 2, etc.), and the atomic Rydberg constant, RA, is given by RA = R∞

( 1 + me MA

)− 1

. (I-14) a) The Rydberg constant for an infinite mass is

R∞ =^2 π

(^2) me e 4 c h^3 = 109, 737 .31 cm−^1 , (I-15) where e = 4. 80325 × 10 −^10 esu is the electron charge in cgs units.

b) In atomic mass units (amu), the electron mass is me =

5. 48597 × 10 −^4 amu whereas the atomic mass, MA, can be found on a periodic table (see also Table I-1).

c) Eq. (I-13) can also be expressed in wavelengths (vacuum) by the following 1 λmn

= RA Z^2

n^2

m^2

)

. (I-16)