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2.5.5 Modern Physics 3

2.5.5.1 Overview

The last part of Modern Physics here (Modern Physics 3 and 4) would range to Schrodinger Equation and the Uncertainty Principle.
Modern Physics 3 derives the basic form of Schrodinger Equation.

2.5.5.2 Details

2.5.5.2.1 Schrödinger Equation in 1926 CE

2.5.5.2.1.1 Background

De Broglie claimed matter travels as a wave with the wave length of λ=h/(m*v). However, details of the shapes of the waves were unclear.
On the other hand, Einstein claimed the energy of a photon is E=h*f.
Other than that, according to Newtonian Mechanics, energy of an object is E=m*v^2/2+V(x).
(V(x): potential energy depending on "x"; x: a point of location corresponding to "x axis") (For example, potential energy is accumulated in a vase raised on a shelf against the force of gravity. When the vase falls, the potential energy changes into kinetic energy of the falling vase. Similarly, potential energy is accumulated in a string and a bow pulled against the force of the string. When the string is released, the potential energy changes into kinetic energy of the arrow. Thus potential energy is accumulated when something is moved against force.)

2.5.5.2.1.2 Schrodinger Equation

Schrodinger sought the shapes of de Broglie's waves.
Schrodinger derived Equation relating to de Broglie's wave.
The following is a simple form of the Schrodinger Equation, time-dependent equation (nonstationary wave) on a free particle as an introductory explanation. (The Schrodinger Equation states a part of conditions that wave functions (ψ) should comply with.)

i * hbar * ∂ψ(x, t)/∂t=-hbar^2/(2*m) * ∂^2ψ(x, t)/∂x^2 + V(x)ψ(x, t)

* "Free Particle in Wikipedia" http://en.wikipedia.org/wiki/Free_particle#Non-Relativistic_Quantum_Free_Particle
* "Schrodinger Equation in Wikipedia" http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation#Time-dependent_equation
" i " is the imaginary unit defined as " i ^2 = -1 ."
* "Imaginary Unit in Wikipedia" http://en.wikipedia.org/wiki/Imaginary_unit
" * " tentatively employed here is the multiplication sign occasionally employed in mathematics instead of "x" to avoid confusion between "ex" of the 24th letter of alphabet.
" hbar " tentatively employed here is "reduced Planck Constant" or "h-bar," commonly described as "", defined as "Reduced Planck Constant" is h/(2*π). As 2*π implies, h-bar is the quantum related to angular momentum.
" ψ (psi)" represents a numerical value of dimensionless quantity in a sense meaning the height or the thickness of the wave at the point of location "x" and at the point of time "t". (Since the concept of de Broglie's wave itself is unclear, the concept of ψ is yet unclear.) On the other hand, ψ is associated with density or concentration of electrons presented in de Broglie's wave. The numerical value of ψ varies depending on "x" and "t". (In addition, the numerical value of ψ varies depending on "V" as the equation implies.)
" x " means a point of location along "x axis." The direction of " x " could tentatively be presumed, for example to the right.
" t " means a point of time along "t axis."
" ∂ " is "partial derivative symbol," "round d," or "del."
" ∂ψ/∂t " is "partial derivative of ψ with respect to t." It means the degree of inclination of ψ against the direction of "t axis." Then ∂ψ/∂t means the amount of increase of ∂ψ per time (for example per "sec") at the "t" point of time.

2.5.5.2.1.3 Shape of Orbits (Shape of Waves) around a Nucleus

Consequently, the shapes of orbits, waves, or ψ (wave functions) around a nucleus are like below according to conditions around the nucleus, as mentioned above involving Electron Configuration.

Attribution: http://en.wikipedia.org/wiki/File:Electron_orbitals.svg

2.5.5.2.1.4 Derivation of Schrodinger Equation

The first stage of the derivation of the Schrodinger Equation (time-dependent (nonstationary wave) on a free particle) (a part of conditions that wave functions (ψ) should comply with) is presuming a simple wave function as an example. A simple example of a function describing a simple wave would be a "cosine curve" of trigonometric (and complete) functions. A cosine curve could be illustrated as follows.
Orange dots show edges of one side of the wave.
Blue dots show edges of the other side of the wave.
(The vertical axis is tentatively described as "Re.")

Then the 1st presumption of ψ as an example of a time-dependent wave in reference to a free particle (a free electron) would be " ψ(x, t)=A*cos 2*π*(x/λ-f*t) ."
" A " is a certain numerical value like an amplitude of a swing.
The cosine curve would proceed to the right (to the direction of "x") like "wave 1," "wave 2," "wave 3," and "wave 4" as time ("t") passes as illustrated as follows.


However, trigonometric curves such as cosine curves have 3 problems in this case.
The 1st problem is that generally shapes of waves are not exclusively cosine curves or sine curves. It means that validity of the presumption employing a cosine curve as an example should be examined considering various possible shapes of waves. However, "orthogonality" of trigonometric functions such as cosine curves and approximation through "sum of series of orthogonal functions' set" such as "Fourier Series" solves this problem.
As Fourier Series tells, any shape of functions could be approximated by sum of trigonometric functions within a certain range.
Just for example, a function F(x) (a square wave) coud be approximated as
" F(x)= 0*1 + 0*cos x + (4/π)*cos (x-π/2)
+ 0*cos (2*x) + 0*cos (2*(x-π/2))
+ 0*cos (3*x) + (4/(3*π))*cos (3*(x-π/2))
+ 0*cos (4*x) + 0*cos (4*(x-π/2))
+ 0*cos (5*x) + (4/(5*π))*cos (5*(x-π/2)) + ... ."
*In this case, "L" (wave length) in the following website is presumed 1.
*In addition, although both "cos" and "sin" are commonly employed in Fourier Series, "sin x" are replaced here with "cos (x-π/2)" for a simple explanation, since "sin x" = "cos (x-π/2)."
* "Mathworld Fourier Series Square Wave" http://mathworld.wolfram.com/FourierSeriesSquareWave.html
* "Mathworld Fourier Series" http://mathworld.wolfram.com/FourierSeries.html
(Coefficients and functions, such as " 0*cos x ," " (4/π)*cos (x-π/2) ," and " 0*cos (2*x) " follow nearly infinitely for perfect approximation.)
* "Fourier Series in Wikipedia" http://en.wikipedia.org/wiki/Fourier_series
* "Approximation to a Square Wave in Wikipedia" http://en.wikipedia.org/wiki/File:Fourier_Series.svg
This property (ability of universal approximation) is called completeness, to be precise.
Then even if actual wave functions (ψ) (true wave functions) are irrelevant to the very cosine curves or sine curves, any shapes of wave functions would be interpreted as mere composites of cosine curves or sine curves. Then the presumption employing cosine curves (or sine curves) as an example has no serious problem anyway. In addition, since the derivation here is a mere presumption, rigorous reasoning wouldn't be required. Accuracy of presumptions would be evaluated according to their conformity with reality (experimental results).
(Consequently, as mentioned later, results from the Schrodinger Equation were quite similar to experimental results, and the Schrodinger Equation was recognized quite accurate.)

By the way other than that, the above example of Fourier Series tells that when an array of a series of (the right side) functions is presumed in advance, the left side function could be expressed by an array of coefficients. Specifically, when a series of the right side functions is presumed as
" 1, cos x, cos (x-π/2), cos (2*x), cos (2*(x-π/2)), cos (3*x), cos (3*(x-π/2)), ..., cos (n*x), cos (n*(x-π/2)),...) ,"
the left side function could be expressed by an array of coefficients like
" 0, 0, 4/π, 0, 0, 0, 4/(3*π), 0, 0, 0, 4/(5*π), 0, 0, ... ."
An array of coefficients could be mathematically regarded as "vector" or "matrix." Then any function could be expressed as a "vector" (commonly a column vector) or a "matrix" (commonly a single column matrix).
* "Coordinate Space in Wikipedia" http://en.wikipedia.org/wiki/Coordinate_space
* "Column Vector in Wikipedia" http://en.wikipedia.org/wiki/Column_vector
* "Matrix in Wikipedia" http://en.wikipedia.org/wiki/Matrix_(mathematics)

The 2nd problem of the presumption on ψ employing cosines is that it brings about mathematical difficulty in the further mathematical process.
The 3rd problem of the presumption on ψ employing cosines is that it includes information of neither momentum nor energy. Supposing a swing of a pendulum, the pendulum stops at the edges of the swing with no velocity or momentum. On the other hand, at the middle point, the pendulum shows the maximum velocity and momentum, while the total energy of the pendulum is constant. The presumption employing cosines wouldn't include the information of momentum or the constant energy.
These problems are solved through imaginary number, complex plane, and Euler's Formula.
An imaginary numbers is a number whose square is less than zero. The square root of "-1" is defined as "imaginary unit" represented by " i ." Then the square of i ( i 2 ) is "-1." For example as an imaginary number, "square root of -9" is 3*i.
* "Imaginary Number in Wikipedia" http://en.wikipedia.org/wiki/Imaginary_number
A complex number is a number defined as the sum of an ordinary number (real number) and an imaginary number. For example, " 3.6 -2i " is a complex number. Another example is " -4π +5.1*i ." (The real number of a complex number is called "real part." The imaginary number of a complex number is called "imaginary part.") (Multiplication sign such as " * " and " x " would be employed depending on occasions.)
Other than that, when numbers are represented by alphabets, like " a + i b " and " x + i *y " could be examples.
Complex numbers are commonly explained on "complex plane."
* "Complex Number in Wikipedia" http://en.wikipedia.org/wiki/Complex_numbers
* "Complex Plane in Wikipedia" http://en.wikipedia.org/wiki/Complex_plane
The following would be an example of complex plane.

A complex number " 4 + 3i " (named (1)) is illustrated on a complex plane. The horizontal axis "Re" means real number, the vertical axis "Im" means imaginary number. Then (1) is placed at the 4th scale rightward and 3rd scale upward.
By the way, the size (value) of a complex number is defined as the distance from the origin of the coordinate axes (the radius on the complex plane). In this case, " r " is the size of " 4 + 3 i ." " r " is 5 at a glance.
However, according to Pythagorean Theorem, the square of the diagonal of a rectangle is the sum of the square of the long side and the square of the short side. Then in ordinary cases, it is calculated as the square root of the sum of the square of the both side. However, in this case, the sum of the square of the both side is "4*4 + -3*3 " = 7. Then " r " falls in square root of 7 = 2.646.
* "Pythagorean Theorem in Wikipedia" http://en.wikipedia.org/wiki/Pythagorean_theorem
In this case, the correct way to calculate the size (absolute value) is as follows.
(a) First, presume a complex number inverting the plus or minus sign of the imaginary part. For example on " 4 + 3*i ," presume " 4 - 3*i ." The complex number inverted on the plus or minus of the imaginary part is called a "conjugate complex number."
"Conjugate Complex Number" is sometimes indicated employing an overline and so on.
(b) Second, multiply the conjugate complex number and the original complex number.
( ( 4 - 3*i ) * ( 4 + 3*i ) = 4*4 + 4*3*i -3*i *4 - 3*3*i^2 = 16 - (-9) = 25 ).
(c) Third, extract the square root ( r = square root of 25 = 5 ).
* "Complex Conjugate in Wikipedia" http://en.wikipedia.org/wiki/Complex_conjugate

*By the way just for further introduction to Matrix of quantum mechanics, Matrix is frequently helpful in calculation.
For example, the following is an exercise to calculate the coordinate value of a2 rotated counterclockwise from a1 by π/3.


This exercise is solved employing a Rotation Matrix.
* "Rotation Matrix in Wikipedia" http://en.wikipedia.org/wiki/Rotation_matrix
The following shows a Rotation Matrix operated on a1 (x1, y1).

cos θ -sin θ x1 cos θ * x1 - sin θ * y1
=
sin θ cos θ y1 sin θ * x1 + cos θ * y1

In this case, θ = π/3, x1 = 4, and y1 = 1.
Then,

cos π/3 -sin π/3 4 cos π/3 * 4 - sin π/3 * 1
=
sin π/3 cos π/3 1 sin π/3 * 4 + cos π/3 * 1

Then

0.5 -0.866 4 1.134
=
0.866 0.5 1 3.964

A matrix helps calculation so easy.


Euler's Formula is
e i x = cos x + i * sin x
e: a mathematical constant defined as the sum of the infinite series,
e = 1 + 1/1 + 1/(1*2) + 1/(1*2*3) + 1/(1*2*3*4) + ... , approximately 2.7183.
* "Euler's Formula in Wikipedia" http://en.wikipedia.org/wiki/Euler's_formula
* "E in Wikipedia" http://en.wikipedia.org/wiki/E_(mathematical_constant)
Then if a formula is presumed as " ei * x = a + b * i " (a, b, and x are real numbers), the traces of "a" and "b" depending on variation of "x" would be illustrated as follows.

Then if "x" is converted into an axis, it would be illustrated as follows. ("a" axis is replaced with "Re" representing real number, "b" axis is replaced with "Im" representing imaginary number.)

Extracting "Re" (real number) and "x" from this spiral, a cosine curve is seen as follows.

Extracting "Im" (imaginary number) and "x" from the spiral, a sine curve is seen as follows.
(By the way, partial derivative of eα*x (by "x") is α*eα*x)

Then the following form of ψ(x, t) would be an adequate example of ψ(x, t) to derive the equation for now.
(Variables of functions such as "(x, t)" are generally sometimes left out)
ψ = A*e 2*π* i *(m*v*x - E*t)/h  --------------- Equation (1)

The following equations are derived from Equation (1).
Partially differentiating with respect to "x",
∂ψ/∂x = 2*π* i * m * v * A/h * e 2*π* i *(m*v*x - E*t)/h
= 2*π* i * m * v * ψ/h
= i * m * v * ψ/hbar
As mentioned before in reference to Mass-Energy Equivalence, momentum (p) could be one of primary variables.
Then replacing with p (momentum) as a preparation to introduce operators,
∂ψ/∂x = i * m * v * ψ/hbar = i * p * ψ/hbar
p * ψ = - i * hbar * ∂ψ/∂x
Then "momentum operator - p(ope)" mentioned below could be assumed as follows.
p(ope) = -i * hbar * ∂/∂x ------------------ Equation (2)
On the other hand, partially differentiating Equation (1) with respect to "t",
i * hbar * ∂ψ/∂t = E * ψ
Then "energy operator - E(ope)" mentioned below could be assumed as follows.
E(ope) = i * hbar * ∂/∂t -------------------- Equation (3)

On the other hand, according to Newtonian Mechanics, energy of an object is
E=m*v^2/2+V(x)
Then replacing with p (momentum) (as a preparation to introduce operators),
E=p^2/(2*m)+V(x) ------------------------- Equation (4)
Transforming the equation assumptively following Quantum Mechanics compounding with ψ and changing "E" and "p" into operators,
E(ope)*ψ=p(ope)^2 * ψ/(2*m) + V(x) * ψ ---------------- Equation (5)
(V(x): potential energy depending on "x"; x: a point of location corresponding to "x axis")

In this case, E(ope) (tentatively (ope) is added) is an operator (command) to calculate Energy in relation to wave function ψ. Naturally, E(ope)*ψ=E (energy of ψ).
p(ope) is an operator (command) to calculate momentum in relation to wave function ψ. Naturally, p(ope)*ψ=p=m*v (momentum of ψ).
"An Operator" roughly means a sign to command readers (students) to calculate complying with the process designated by the sign. For example, "+" in "3+5" is an operator commanding the readers (students) to add 3 to 5. For example, "^" in "5^2" is another operator commanding the readers to multiply two "5", 5 x 5.
The concept of "Operator" came from "Operational Calculus" roughly advocated by the physicist Oliver Heaviside. In this concept, "a measured value" is regarded as a composition of "measuring method" and "an object to be measured." For example, "Mr. Smith's blood pressure (a measured value)" is a composition of "sphygmomanometry (measuring method)" and "Mr. Smith (an object)." "Mr. Smith's body weight" is a composition of "body weight measuring method" and "Mr. Smith."
"Momentum of an electron" is a composition of "Momentum calculating method" and "an electron," while "Momentum calculating method" is "momentum operator ( p(ope) )" and "an electron" is "ψ". Then "momentum (p)" is replaced with "p(ope)*ψ", then "p = p(ope)*ψ." "Energy (E)" is replaced with "energy operator (E(ope))" and "ψ", then "E = E(ope)*ψ."
* "Operational Calculus in Wikipedia" http://en.wikipedia.org/wiki/Operational_calculus
"Operators" are signs, while operators are commonly employed with variables to define a function. For example, if a function " F " is defined as " F(x) = x+2 ," " + " is an operator and " x " is a variable. Functions consist of operators (signs or commands to calculate) and variables. Whether "functions (simplified functions leaving out (x, t) and so on)," "operators (signs or commands to calculate)," or "variables" should be wisely distinguished.
Operators could be distinguished through tentative signs such as (ope), while signs (such as "(ope)") tend to be left out.

(On the other hand, it shoud be noted that the Schrodinger Equation is besed on E=m*v^2/2, Newtonian Mechanics, excluding the Theory of Relativity.)

Then the general Schrodinger Equation, time-dependent (including nonstationary waves and stationary waves) on a free particle, is derived as follows substituting Equation (2) and Equation (3) for Equation (5).

i * hbar * ∂ψ(x, t)/∂t=-hbar^2/(2*m) * ∂^2 ψ(x, t)/∂x^2 + V(x) * ψ(x, t) ----- Equation (6)

However, since this general Schrodinger Equation is general, specific wave functions wouldn't be derived from this Equation. Additional conditions such as shape of potential energy function ( V(x) ) and/or stationarity of waves should be designated as examplified below to find specific wave functions.

2.5.5.2.1.5 Klein-Gordon Equation in 1926 CE

In relation to Schrodinger Equation, Klein- Gordon Equation was presented. As mentioned before, Einstein presented Mass-Energy Equivalence as follows.
E^2=m^2*c^4 + c^2*p^2 --------- Equation (7)
It was assumptively transformed following Quantum Mechanics compounding with ψ and changing "E" and "p" into operators as follows.
E(ope)^2 * ψ = m^2 * c^4 * ψ + c^2 * p(ope)^2 * ψ ---------- Equation (8)
Then substituting Equation (2) and (3) for Equation (8), Klein-Gordon Equation was derived as follows.

hbar^2 * ∂^2 ψ/∂t^2 = - c^2 * hbar^2 * ∂^2 ψ/∂x^2 + m^2 * c^4 * ψ ------- Equation (9)

It follows the Special Theory of Relativity, but the equation couldn't be resolved anymore. The equation was quite complicated and things seemed tangled.

* "Klein-Gordon Equation in Wikipedia" http://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation

2.5.5.2.1.6 Clear Derivation from Perfect Elements

By the way, as mentioned above, neither sine curves nor cosine curves would result in clear derivation (conclusion), but an Euler's Formula resulted in clear derivation (conclusion). Then it should be noted that, in a sense, perfect elements such as Euler's Formula would result in clear mathematical resolution, in contrast to that, imperfect elements such as sine curves and cosine curves would result in difficulties in mathematical analysis. In other words, clear mathematical analysis implies essential perfectness of the elements such as Euler's Formula. In contrast to that, difficulty in mathematical analysis implies imperfection of the elements such as sine curves and cosine curves.






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