* Basic laws of Heat – thermal behavior*:

Definition of Energy:

Operational : "

*The ability to do work*" - e.g. W = F x d (force acting in a direction x displacement in that direction)

Ideal definition: (Noether's):

*“Energy is that quantity that’s conserved because of time-displacement symmetry”*

The last segment, “time-displacement symmetry” refers to the constancy of physical laws in time. Time goes on, but the laws of physics retain a constancy of their properties within it.

Law of conservation of energy-mass:

“

*The total amount of mass-energy in the universe in a system remains constant*.”

This includes as a generalization the first law of thermodynamics, e.g. that internal (heat) energy is conserved, i.e. if two bodies are in thermal contact – and at different temperatures- the cooler body will have heat energy transferred to it, while the hotter will LOSE heat energy.

The Second law of thermodynamics (entropy law) is simply stated as:

“

*The entropy (degree of disorder of a system, usually denoted by the symbol S) increases in all natural processes*”

There are two subsidiary statements of it:

**The Kelvin -Planck statement**:

*It is impossible to construct a heat engine, operating in a cycle, which produces no other effect than absorption of thermal energy from a hot reservoir and the performance of an equal amount of work.*

**II): The Clausius statement**:

*It is impossible to construct a cyclical machine that produces no other effect than to transfer heat continuously from one body to another at higher temperature.***Basic
laws of Electricity and Magnetism: **

Coulomb’s Law: Any 2 charged particles (e.g. electrons, ions) attract (or
repel) each other with a force inversely proportional the square of the
distance between them and directly proportional to the product of the charges
(Q, q)

F = k Qq/ r^{2}

where k is a constant, Q, q the charges, and r the separation. Note the similar
mathematical form to the universal law of gravitation. The difference is that
the latter is ONLY attractive, while the Coulomb interaction may also be
repulsive. (E.g. two like charges will always repel)

Gauss’ Law: The net number of electric field lines passing through a surface
that encloses a net electric charge is proportional to the charge enclosed
within the surface.

* Other Electric –magnetic laws*:

*“There are NO magnetic monopoles. All magnetic field lines must end on one or other of two poles.”*

Changes in magnetic flux always produce an induced electric current (Faraday’s law)

Moving electric charges in a closed circuit or loop give rise to a magnetic field (Ampere’s law)

From Maxwell’s E-M equations:

“

*At every instant, the ratio of the electric field magnitude to the magnetic field magnitude equals the speed of light, c” viz.*

*(**ú**
E**ú** /**ú** B**ú** = c )*

Generalized law for E-M waves arising from Maxwell’s equations, laws:

“Electro-magnetic waves are generated by accelerating charges and consist of
oscillating electric and magnetic fields which are at right angles to each
other and also at right angles to the direction of wave propagation.”

In general, Maxwell’s equations will be expressed:

i) **Ñ**** X H **= **J **+ ¶**D** / ¶ t
(A current density **J** arises from a magnetic field)

ii)
**Ñ**** X E
**= **-** ¶**B** / ¶ t
(A magnetic field can arise from an electric field)

iii)
**Ñ**** ****·**** B **= **0 **(There are no magnetic monopoles)

iv)
**Ñ**** ****·**** D **= r ** **(Charges are conserved)

In addition, there are three “constitutive relations” that allow any of the above vectors to be re-cast in slightly different forms:

v) **D**
= e **E**

vi) **B** =
m **H**** **

vii)**
J **= s **E**

** **

In
the equations above, **H **represents the *magnetic field intensity*, **B** is the *magnetic induction*, **E** the *electric field intensity*, **D** the *displacement current*, and **J **is the *current density*. The constants, e and
m, denote the permittivity
and the magnetic permeability – each for media. In vacuo, the constants used
are: e _{0} and m _{0} and the speed of light can be
expressed: c = 1/ Öe _{0} Ö m _{0} .

*:*

**Basic laws of quantum mechanics**Whenever electrons change position (energy levels) in an atom, energy is given off in a discrete packet such that:

E = h n

where h is the Planck constant, and f is the frequency of the emitted light (photon) corresponding to the difference of energies in the levels.

This may also be written: E = hc/ l

where l is the wavelength

Every material particle has associated with it a

*de Broglie wave*with a wavelength

l = h/ mv

where h is Planck’s constant, and m the mass, v the velocity.

All atoms represent systems that can be described in terms of probability waves, such that these waves disclose the probability of where the constituent electrons are at any given time. The core equation foundational to describe these waves is the Schrodinger Equation:

*:*

**Basic principles of quantum mechanics**Whenever electrons change position (energy levels) in an atom, energy is given off in a discrete packet such that:

E = h n

where h is the Planck constant, and f is the frequency of the emitted light (photon) corresponding to the difference of energies in the levels.

This may also be written: E = hc/ l

where l is the wavelength

Every material particle has associated with it a de Broglie wave with a wavelength

l = h/ mv = h/ p

where h is Planck’s constant, and m the mass, v the velocity.

**:**

*Postulates of quantum mechanics***Postulate 1**. The state of a quantum mechanical system is completely
specified by a function that depends on the coordinates of
the particle(s) and on time. This function, called the *wave function *or state
function, has the important property that:

y (r,t) y *(r,t) dV

is the probability that the particle lies in the volume element dV.

The wavefunction must satisfy certain
mathematical conditions because of this probabilistic interpretation. For the
case of a single particle, the probability of finding it *somewhere* is 1, so that we have the normalization condition:

P _{aa’ } = ò ^{a’}_{a}_{ } ‖y‖^{2 } dx = 1

All atoms represent systems that can be described in terms of probability waves, such that these waves disclose the probability of where the constituent electrons are at any given time. The

*Schrodinger equation*embodies the foundational description and principles to encompass this subatomic behavior.

An analogous approach can be used to obtain the probability density (e,g, for electron in 1s state) for the hydrogen electron (in 3 dimensions):

**P** = ½y (1s) y (1s) *½

yielding:

**The n = 1 electron orbital for hydrogen**

As well as the energy quantization equation for the spectral lines. e.g.

E
= -** **ħ** ^{2 }/ **2m [a

^{2}- 2a/ r]

**-**e

^{2}/ r

**Postulate 2**. To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum mechanics.

For example there are 3 angular momentum operators for each of three Cartesian dimensions, x, y, z, e.g.

**= -i h [y (¶/¶x ) – x (¶/¶y)]**

_{x op }**= -i h [z (¶/¶x ) – x (¶/¶z)]**

_{y op }**= -i h [x (¶/¶y ) – y (¶/¶x)]**

_{z op }**Postulate 3**. In any measurement of the observable associated
with operator **A^**, the only values
that will ever be observed are the eigenvalues a, which satisfy the
eigenvalue equation:

**A^**y = a y

This postulate captures the central point of
quantum mechanics--the values of dynamical variables can be quantized (although
it is still possible to have a continuum of eigenvalues in the case of unbound
states). If the system is in an eigenstate of **A^ ** with an eigenvalue a then any measurement of the quantity A will yield a.

* Heisenberg Uncertainty Principle*:

Given two canonically conjugate variables (e.g. energy and time, position and momentum), they cannot both be measured to the same precision at the same time.

* Pauli Exclusion
Principle*:

*“*No 2 electrons in an atom can ever be in the same quantum state, that is – no two electrons in the same atom can have the same exact set of quantum numbers (n, l, m(s) or ml)”

*A summary of the simpler Bohr model for atomic applications is found in the flow chart below:*

*ATOMIC SUMMARY FLOW CHART: To find(a) Total energy of hydrogen atom, b) radii of allowed orbits, c) Allowed energies of atom for different stationary electron states*

**Einstein’s mass-energy
equation**:
E = mc^{2}

*“In any fusion or fission
reaction, the total rest mass of the products is less than the rest mass of the
reactants – the change (decrease) in rest mass appearing as energy released in
the reaction.”*

Thus: D E (change in energy) = [m_{R}
- m_{p}] c^{2 }

where the bracketed quantity on the right side is the difference in rest masses
between reactants and products.

* Special relativity*:

*:*

__Postulates__**1) The laws of
physics are the same in all inertial reference frames.**

**I.e. "All inertial observers are equivalent."**

**2) The velocity of light, c, is independent of the state of motion of the source and the observer.
**

This implies:

“*the speed of light
has the same value in all inertial reference frames*”

Thus, in no inertial reference frame can any material object exceed the speed of light.

Time dilation:

**All moving clocks run
slower relative to an identical clock in a stationary frame.**

Finally, the “fourth dimension” is NOT a basic law. Rather it arises out of the
Principle of Relativity by virtue of
referencing all physical actions, laws in terms of *time* as well as the three dimensions of space (x, y, z).

In fact, in principle, given the four dimensions x, y, z and t - ANY *ONE *could be “*the fourth
dimension*”! (The order of choice is not important, what’s important is that
four dimensions are required to specify and follow physical laws between
difference reference frames).

* The Interval*:

Then the space-time intervals (r and r') will be such that:

^{2}
= x^{2} + y^{2} + z^{2} - c^{2} t^{2}

r’^{2}
= x’^{2} + y’^{2} + z’^{2} = c^{2} t’^{2}

*And also:*

x^{2}
+ y^{2} + z^{2} - c^{2}
t^{2} = x’^{2} + y’^{2}
+ z’^{2} - c^{2} t’^{2}

**The Inertia of Energy**:

This
is embodied in Einstein's famous equation:

E = m c^{2}

which is more accurately posed as:

E = (D m) c^{2}

where D m is the "mass
defect" or difference, say in a nuclear reaction, and c is the velocity of
light.

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