Disclaimer: This is Totally Untrue.
2.5.6 Modern Physics 4
From here, as the last part of Modern Physics,
Schrodinger Equation is practically reformed and the Uncertainty Principle is introduced.
22.214.171.124.1 Practical Schrödinger Equation in 1926 CE
126.96.36.199.1.1 Stationary Waves like Stepping-Stones in Potential Wells
From here, in relation to Schrodinger Equation,
additional conditions are presumed to find specific wave functions.
V(x) means potential energy varying depending on location (x).
The following would be an example of V(x).
If a nucleus with a proton locates at the origin of the coordinate "x" axis (at the center),
an electron rather distant would be attracted to the proton.
Then the potential energy would be reduced approaching the proton.
However, because the nucleus and the electron would not overlap, the potential energy
near the nucleus (at the center) would rise.
The shape of the wave would vary depending on the shape of the potential energy, V(x).
On the other hand, time-dependent Schrodinger Equation
mentioned above includes nonstationary waves (in addition to stationary waves).
However, interesting waves are stationary waves.
(Since stationary waves are not closely associated with time passage,
they are called "time-independent.")
Examples of stationary waves would be as follows.
Stationary waves would be tentatively represented as
" ψ(x, t) = ψst(x)*sw(t) ,"
where ψst(x) here means a function representing shape of
a stationary wave,
sw(t) here means a function representing degree of
a swing of a stationary wave.
Hight (degree of swing) of stationary waves would vary depending on time.
But the basic shape (except the height) wouldn't vary depending on time
(naturally vary depending on location (x)).
Then the wave function would be separated
into the factor of shape depending on location " ψst(x) "
and the factor of height (degree of swing) depending on time " sw(t) ."
In other words, a stationary wave is the mathematical product of
"the factor of shape depending on location, ψst(x)" and
"the factor of height depending on time, sw(t)."
Then " ψ(x, t) = ψst(x)*sw(t) " as far as stationary waves.
The next simple presumption is that the shape of the
potential energy is like a square well. (The height of the wall is presumed infinite.)
"Particle in a Box in Wikipedia"
Summarizing these presumptions,
(-hbar^2/(2*m)*∂^2/∂x^2 + V(x))ψst(x)sw(t) = i hbar*∂ψst(x)sw(t)/∂t
(-hbar^2/(2*m)*∂^2 ψst(x)sw(t)/∂x^2 + V(x)ψst(x)sw(t) = i hbar*∂ψst(x)sw(t)/∂t
-sw(t)hbar^2/(2*m)*∂^2 ψst(x)/∂x^2 + V(x)ψst(x)sw(t) = ψst(x)* i hbar*∂sw(t)/∂t
-1/ψst(x)*hbar^2/(2*m)*∂^2 ψst(x)/∂x^2 + V(x) = 1/sw(t) * i *hbar*∂sw(t)/∂t
The left side depending on x equivalent to the right side depending on t means they are constant.
Then the constant is tentatively called here " Econst ."
The left side = Econst is transformed into
(-hbar^2/(2*m)*∂^2 /∂x^2 + V(x) )ψst(x) = Econst*ψst(x)
Otherwise, since the height of the wall is presumed infinite,
Then if 0≦x≦L ,
-hbar^2/(2*m)*∂^2 ψst(x)/∂x^2 = Econst*ψst(x)
(-hbar^2/(2*m)*∂^2 /∂x^2 + ∞ )ψst(x) = Econst*ψst(x)
This latter ("otherwise": except 0≦x≦L) means ψst(x)=0
∂^2 ψst(x)/∂x^2 = -2*m*Econst*ψst(x) /hbar^2
Here, "k" is tentatively defined as SQRT(2*m*Econst/hbar^2)=k
∂^2 ψst(x)/∂x^2 = -k^2*ψst(x)
In this case, ψst(x)=eαx (α: a complex number) can be a solution.
(α^2 + k^2)=0
α = ± i *k
ψst(x) = e i *k*x or e -i *k*x
ψst(x) = A*e -i *k*x + B*e i *k*x
= A*(cos(kx) - i *sin(kx)) + B*(cos(kx) + i *sin(kx))
Here, italic A and B (different from A and B) are
defined as follows.
ψst(x)=A*sin(kx) + B*cos(kx)
(k: wave number)
The edges of stationary waves should be zero.
A*sin0 + B*cos0 = 0
B = 0
On the other hand,
ψst(L) = 0
ψst(L)=A*sin(kL) + B*cos(kL)=0
kL=nπ (n=1, 2, 3, 4, ...)
k*x = n*π/L
ψst(x)= A * sin (π*x/L), A * sin (2*π*x/L),
A * sin (3*π*x/L),
A * sin (4*π*x/L), ...
ψ(x, t)= A * sin (π*x/L), A * sin (2*π*x/L),
A * sin (3*π*x/L),
A * sin (4*π*x/L), ...
Thus, the functions vary like stepping-stones depending on "n".
By the way, the Energy Eigenvalue varying depending on "n" in this case is
En = n^2*hbar^2*π^2/(2*m*L^2)
Summarizing the functions and Energy, they would be illustrated as follows.
188.8.131.52.1.2 Improvement of the Schrodinger Equation 1 (Choice of Variables and Coordinate System)
By the way, choices of variables and coordinate system are proposed for rational analysis.
One choice is Lagrangian Form and the other is Hamiltonian Form.
Typical principle variables in Newtonian mechanics are "t (time),"
"x, y, z (location or distance about 3-dimensional space)," and "m (mass)."
However, these variables wouldn't match further mathematical analysis.
Choice of variables should be considered.
Then "Lagrangian Form" or "Lagrangian Mechanics"
proposes to choose 2 variables.
One variable is " q " representing location or distance
under "Generalized Coordinate System" (instead of x, y, z).
"Generalized Coordinate System" is, in short,
a convenient (selfish) coordinate system for each analyst (physicist)
to indicate location or distance instead of x, y, z.
In other words, "x, y, z coordinate system" of Newtonian mechanics can't
clearly deal with coordinate transformation.
For example, if the coordinate system is transformed from x, y, z to the polar coordinate system,
"force" would be described as " force = - d(V + (m*r^2*(∂θ/∂t)^2/(2*m*r^2))/dr ,"
rather in a complicated formula.
Then Generalized Coordinate System is proposed to avoid such complications.
Location or distance is indicated by " q " implying that the coordinate system adopted is
Generalized Coordinate System.
The other variable is
" " (" ● (dot)" crowned " q "),
representing velocity (generalized velocity) based on "q " (generalized coordinate system).
(" q' " might be tentatively employed here instead
of " ● " crowned " q " because of the difficulty of the letter type.)
"Generalized Coordinates in Wikipedia"
"Lagrangian Mechanics in Wikipedia"
The other proposal is "Hamiltonian Form" or "Hamiltonian Mechanics."
Hamiltonian Form or Hamiltonian Mechanics also proposes to choose 2 variables.
One variable is " q " representing location or distance
under "generalized coordinate system," the same variable as Lagrangian Mechanics.
The other variable is " p " representing momentum (generalized momentum).
"Hamiltonian Mechanics in Wikipedia"
Since Lagrangian Form and Hamiltonian Form are replace of variables from
Newtonian mechanics for mathematical analysis,
Newton's Second Law could be transformed into
"Lagrangian Equation of Motion" and "Hamiltonian Equation of Motion."
In addition by the way, Lagrangian Mechanics proposes
an index named "Lagrangian"
commonly represented by " L ".
Lagrangian (L) is defined as " Lagrangian = T - V ."
T is kinetic energy and V is potential energy.
According to Newtonian mechanics, Energy (E) is
defined as " E = T (kinetic energy) + V (potential energy) ."
In contrast to that, " Lagrangian = T (kinetic energy) - V (potential energy) ."
The meaning of Lagrangian would be like
"concentrated or summarized information related to the state of motion."
Other than that for example, Newton's Second Law could be transformed employing Lagrangian as
" d(∂L/∂x)/dt = ∂L/∂x " to be called "Lagrangian Equation of Motion."
Lagrangian Equation of Motion would neither change its form nor fall in complicated form
regardless of coordinate transformation.
It implies essential perfectness of Lagrangian Mechanics.
The context of Lagrangian would be as follows.
Firstly, essential concept lurking in Newtonian mechanics like Brachistochrone curves
Then "the Principle of Least Action" likely associated with economical (efficient) trajectories was found.
It means "Action" represented by "S" is kept to a minimum in Newtonian mechanics.
"Brachistochrone Curve in Wikipedia"
"Principle of Least Action in Wikipedia"
S is defined as the integral of the Lagrangian between two instants of time (for example, t1 and t2).
Consequently, Lagrangian = T (kinetic energy) minus V (potential energy) = T -V.
The legitimacy of Lagrangian would be verified as follows.
Supposing a force (to the right) on a ball laid on a slant of
potential energy varying depending on x ( V(x) ), the force (to the right)
corresponds to the slant of the potential energy.
For example for easy understanding, supposing the counter force (to the left)
keeping the ball stationary, it corresponds to
the slant of the potential energy.
(Since it is a simplified example for explanation, vertical force should be ignored.)
" force = m*a = m*d^2x/dt^2 = -dV(x)/dt "
(Since motion depends on time, "x" varies depending on "t", "x" can be replaced with "x(t)," to be precise.)
On the other hand, "Action" (S) is consequently defined as follows.
S= integral of ( 1/2*m*(dx/dt)^2 - V(x) ) over t1-t2
If an original trajectory (path) of an object is presumed employing "x" and "t" (the function
presumed here is "x(t)") and
a new trajectory (the function presumed here is "xnew(t)")
with small differences (small changes)
from the original trajectory
small differences related to S (" δS ") and small differences (small changes) related to
x (" δx ") are presumed as follows. (Since δx varies depending on t,
δx can be replaced with δx(t), to be precise.)
δS = integral of ( (1/2*m*(d(x+δx)/dt)^2 - V(x+δx))
- (1/2*m*(dx/dt)^2 - V(x)) ) over t1-t2
(S and δS vary depending on trajectories of objects represented by
functions related to the trajectories, x(t). ("x(t)" and "t" represent trajectories.))
Decomposing the 1st term and ignoring δx^2 (since δx^2 is extremely small),
δS = integral of (m*dx/dt*dδx/dt - δx*∂V(x)/∂x) over t1-t2
Integrating by parts in reference to the 1st term,
δS = m*dx/dt*(δx(t2)-δx(t1)) - integral of (m*d^2x/dt^2*δx - δx*∂V(x)/∂x) over t1-t2
"Integration by Parts in Wikipedia"
Since t1 and t2 are edges (boundaries) of the integration ("surface term" in mathematics),
δx(t2) and δx(t1) are presumed to be zero.
"Surface Term in Physics Forums"
δS = integral of δx(-m*d^2x/dt^2 - ∂V(x)/∂x) over t1-t2
Shape of δx(t) is unknown.
However, if -m*d^2x/dt^2-∂V(x)/∂x = 0, δS = 0.
It means under the law
" force = m*a = m*d^2x/dt^2 = -dV(x)/dt " mentioned above,
"Action" (S) defined above presuming Lagrangian to be
"kinetic energy - potential energy" is kept to a minimum.
Then the essence of Newtonian mechanics is
rather the Principle of Least Action.
On the other hand, Hamiltonian Mechanics
proposes an operator (command to calculate) named
"Hamiltonian" commonly represented
by " ".
Operators are commonly crowned by " ^ " to be recognized as operators,
while tentatively "(ope)" might be added here like " H(ope) " to indicate operators.
Other than that,
as mentioned above,
Matrices are convenient tools for calculation.
Matrices are other examples of operators.
Hamiltonian is an operator (sign or command) to calculate "energy."
(In addition, Hamiltonian operator implies that the Hamiltonian Form (main variables are " q " and " p ")
is adopted in reference to the formula.)
"Hamiltonian operator" ( H(ope) ) is defined in the form of function
as " H(ope)(on variables) = T (kinetic energy) + V (potential energy) = energy ."
( T and V would be functions.)
Since Hamiltonian is the operator (command) to calculate energy,
operating Hamiltonian on the object naturally results in energy.
(Other than that, main variables should be " q " and " p ", since Hamiltonian is based on Hamiltonian Form.)
For example, according to Newtonian mechanics, " energy = E = m*v^2/2 + V ."
It could be transformed as " energy = E = H(ope)(q, p) = p^2/(2*m) + V ."
On the other hand, according to the Schrodinger Equation related to a time-dependent free particle,
energy of a particle is interpreted as
" energy = Eψ = m*v^2 * ψ/2 + V(x) * ψ =
i * hbar * ∂ψ(x, t)/∂t=-hbar^2/(2*m) * ∂^2ψ(x, t)/∂x^2 + V(x)*ψ(x, t) ."
Then in the Hamiltonian Form, " energy = H(ope) ψ ."
Then Hamiltonian operator itself related to the Schrodinger Equation of a time-dependent
free particle is
" H(ope) = i * hbar * ∂/∂t=-hbar^2/(2*m) * ∂^2/∂x^2 + V(x) " separating ψ(x, t).
Hamiltonian is merely a sign (or a command) to calculate energy
(mostly based on " q " and " p " for simple mathematical analysis).
Then the specific contents (the way of calculation) of Hamiltonian
operators differ depending on the
184.108.40.206.1.3 Improvement of the Schrodinger Equation 2
The Schrodinger Equation above is then improved aiming at waves of electrons around a hydrogen nucleus.
Furthermore, the Schrodinger Equation is converted to Spherical Polar
Coordinate System to be analized around the nucleus,
employing " r " (radial distance), " θ " (polar angle),
and " φ " (azimuthal angle).
"Spherical Coordinate System in Wikipedia"
"Hyperphysics Spherical Polar Coordinates"
(By the way, due to the definition of θ and φ, "z" axis is positioned as
the simplest axis for analysis.
That's why the "z" axis later enters as the special axis
in reference to (assumed)
(It should be noted that commonly in physics,
"polar angle" is represented by θ, "azimuthal angle" is represented by φ.
In contrast, commonly in mathematics, "polar angle" is
represented by φ, "azimuthal angle" is represented by θ.)
Then, 3 or 4 new functions are presumed, R(r), Y(θ, φ), Θ(θ), and Φ(φ).
R(r) is called Radial Equation associated with principle quantum number
(" n "). ψ(r, θ, φ) is once presumed (separated) as "ψ(r, θ, φ)=R(r)*Y(θ, φ)."
Y(θ, φ) is called "Spherical Harmonics," since the factor of radius is separated.
Then Y(θ, φ) is presumed (separated) as "Y(θ, φ)=Θ(θ)*Φ(φ)" in
accordance with Variable Separation Method.
"Separation of Variables in Wikipedia"
The further analysis leading to Electron Configuration would be as follows.
"Hydrogen-like Atom in Wikipedia"
"Particle in a Spherical Symmetric Potential in Wikipedia"
Θ(θ) is associated with azimuthal quantum number (" l ") and
(assumed) orbital angular momentum (" L "), "Orbital Angular Momentum Operator" to be more precise.
(It should also be noted that although θ here represents "polar angle" as in physics,
it is associated with azimuthal quantum number.)
Φ(φ) is associated with magnetic quantum number
(" m " or more precisely " ml ") and the "z-component of
the (assumed) orbital angular momentum"
(" Lz "). (" L " consists of Lx, Ly, and Lz. L^2=Lx^2 + Ly^2 + Lz^2.)
Consequently, results derived from
the Schrodinger Equation were quite similar to experimental results,
then the Schrodinger Equation was recognized to be a quite
220.127.116.11.1.4 Perturbation Theory for Approximate Calculations
On the other hand, the above would be rather simple examples
consisting of 1 or a few particles and
simple potential energy.
However, realistic circumstances would be far more complicated than that.
The complicated circumstances wouldn't be adequately solved by strict algebraic ways.
Then Schrodinger proposed Perturbation Theory Calculation
for approximate calculations.
"Perturbation Theory in Wikipedia"
18.104.22.168.1.5 Assumed Angular Momentum
Angular momentum was introduced to quantum
mechanics as an analogue of Newtonian Mechanics.
However, "angular momentum" in quantum
mechanics is a mere assumption.
Electrons would not be rotating around the nucleus in reality.
The meaning of ψ or ψ^2 is rather interpreted like clouds of
obscure electrons (or probability of finding electrons).
Then "Angular Momentum (Vector) Operator" would be
more precise instead of "Angular Momentum."
"Angular Momentum Operator" is a kind of Vector and
a vector, an array of numbers, is an analogue
of a matrix. Then "Angular Momentum (Vector) Operator" emerges like in
"Lψ" and "L2 Y(θ, φ)."
(As mentioned above, any function such as ψ and Y can
be described as a column matrix.)
(L2 means operating L twice.)
(" ^ " is frequently placed at the upper side of Operators (alphabet) to specify to be an operator, to be precise.
It might tentatively be replaced here with adding "(ope)" like "L(ope)."
(Angular Momentum Operator is merely an mathematical concept.
Mathematical concepts wouldn't directly connect with real magnetic moment.
There should be more real existence instead of "Operators" behind magnetic moment.
However, details are unclear, Angular Momentum Operator could be a bearable answer for now.)
Consequently based on Schrodinger Equation, L(ope)^2ψ=hbar^2*l*(l+1)ψ.
Then, L(ope)^2=hbar^2*l*(l+1). "size of L(ope)"=hbar*SQRT(l*(l+1))
"Azimuthal Quantum Number in Wikipedia"
L(ope) consists of Lx, Ly, and Lz, as L(ope)^2=Lx(ope)^2 + Ly(ope)^2 + Lz(ope)^2.
"size of Lz"=hbar*ml. ("ml" is Magnetic Quantum Number, in contrast to
Spin Magnetic Quantum Number "ms.")
"Magnetic Quantum Number in Wikipedia"
Then the "total size of the Orbital Angular Momentum Operator" (L) and
"size of the z-component of the Orbital Angular Momentum Operator" (Lz)
are illustrated as follows.
"Vector Model of Orbital Angular Momentum in Wikipedia"
Among 3 components of L (Lx, Ly, Lz), only 1 component is specified from equation,
while the others are unclear supposedly because of
the Uncertainty Principle.
"Uncertainty Principle in Wikipedia"
The specified component is called as Lz, since z-axis is the
easiest axis for analysis.
This could be the context of the law that "the interval among 'ml' is 1,
the maximum of 'ml' is '+l,' and the minimum of 'ml' is '-l'."
22.214.171.124.2 Born's Probability Distribution Interpretation in 1926 CE
Schrodinger presented Schrodinger Equation from matter
wave, but the meaning of ψ was unclear.
Born claimed ψ^2 means probability distribution of electron's presence.
"Interpretaion of Quantum Mechanics in Wikipedia"
126.96.36.199.3 Pauli's Spin Matrices in 1927 CE
Uhlenbeck and Goudsmit presented electron's spin.
However, the assumed spinning surface speed of an
electron was over some 100 times faster than
the speed of light. Then the assumption of spinning seemed unreal.
On the other hand, Schrodinger Equation didn't include spin of electrons.
Pauli presented states of electron spin summarizing the properties
through 3 matrices (2 x 2 matrices), (assumed) angular momentum along
z-axis, x-axis, and y-axis according with the Schrodinger Equation.
The 3 matrices (σ1, σ2, and σ3)
are called Pauli's Spin Matrices or Pauli Matrices.
Pauli's Spin Matrices, in a broad sense, include the
following identity matrix ( I ).
He presented the assumed angular momentum of electron spin
(size of Angular Momentum Operators related to
electron's proper magnetic moment)
of z-component to be +hbar/2 and -hbar/2,
"Spin Magnetic Quantum Number" (ms) to be +1/2 and -1/2.
(Later, +1/2 (or +hbar/2) is called "upspin" and -1/2 (or -hbar/2) is
("Spin Magnetic Quantum Number" or "Spin Projection
Quantum Number" is
rather commonly called "Spin Quantum Number" or "Spin.")
*It should be noted that electrons would not be spinning in reality,
"upspin" and "downspin" are mere concept, while details are yet unclear.
"Hyperphysics Electron Spin"
"Spin in Wikipedia"
"Pauli Matrices in Wikipedia"
Other than that, Pauli improved the theory of so-called "Spin"
(although electrons would not be spinning in reality).
188.8.131.52.4 Heisenberg's Uncertainty Principle in 1927 CE
Behavior of electrons were considered.
2 aspects of elementary particles (Wave-Particle Duality) were
claimed by Einstein and Bohr.
Heisenberg claimed, through his thought experiment (Gamma-Ray Microscope),
in reference to subatomic particles such as electrons,
the more precisely the position is determined,
the less precisely the momentum is
known, and vice versa.
Specifically, "one can never know with perfect accuracy both of those
two important factors which determine the movement of one
of the smallest particles—its position and its velocity.
It is impossible to determine accurately both the position
and the direction and speed of a particle at the same instant."
The concept was formulated to be
"Degree of Uncertainty of Position" * "Degree of Uncertainty of Momentum" is
not less than "a half of the Reduced Planck Constant" ( Δx*Δp≧hbar/2 )
"Degree of Uncertainty of Time" * "Degree of Uncertainty of Energy"
is not less than "a half of the Reduced Planck Constant" ( Δt*ΔE ≧hbar/2 ).
(" * " is a multiplication sign tentatively employed here to
distinguish from " x ".)
The Uncertainty Principle includes 2 meanings, "measuring limit"
and "realistic intrinsic property of particles (or waves)."
As Heisenberg's claim came from the thought experiment
of the Gamma-Ray Microscope,
Heisenberg seems conceived it "measuring limit."
"Uncertainty Principle in
"American Institute of Physics Heisenberg Gamma-Ray Microscope"
"Hyper Physics Uncertainty Principle"
184.108.40.206.5 Lemaitre's Big Bang in 1927 CE
Most nebulae were observed going away from the earth.
The meaning of the observation results was unclear.
Lemaitre derived the Friedmann Equations from Einstein's General Theory of Relativity,
claimed the universe started from a primeval atom and expanded.
Subsequently, Hubble's observation supported Lemaitre's theory.
220.127.116.11 Hund, Gamow, and Condon's Quantum Tunneling in 1928 CE
Molecular mechanism of energy was studied.
Mechanism of Alpha Decay depending on probability was unclear.
Hund's Notion of Tunneling in Double-Well Potential
Hund calculated various ground states of potential energy level,
noted there could be some splitted ground energy states like
"Double-Well" or "W-shaped" and the possibility that
energy state of particles might move from a well to the other well
regardless of the center partitions (walls) of energy
without any additional energy to climb over the partitions (walls).
Hund called it "barrier penetration."
*A representative example of Double-Well Potential
is Nitrogen Inversion in ammonia.
"Science World Tunneling"
"Nitrogen Inversion in Wikipedia"
Gamow and Condon's Alpha Decay Tunneling
Gamow and Condon claimed an account for Alpha Decay depending on probability
extending the Uncertainty Principle to realistic intrinsic property of particles.
Particles such as protons and neutrons are strongly bound to the adjacent regions of nuclei.
It would be compared to a single deep well of a ground state of potential energy.
However, if particles once leave from the nuclei, particles run away from the nuclei.
It means the ground state of potential energy outside of the well decreses depending on distance.
Then the ground state would be like "upside-down W-shaped (M-shaped)" or
"a tall vase with a well-hole at the center."
Gamow and Condon accounted that particles are mostly
hovering at the bottom of the well.
However, particles occasionally appear (emerge) outside of the well
regardless of the barriers or partitions (or walls) of energy.
Once an alpha particle (2 protons and 2 neutrons) emerge outside of the well,
it runs away as an alpha particle resulting in alpha decay.
The emergence occurs based on probability resulting in half-life of alpha decay.
The emergence despite the barriers is based on the Uncertainty Principle
as realistic intrinsic property of particles.
It was called "Quantum Tunneling."
Subsequently, dominion of "existence probability" over
quantum mechanics including Schrodinger Equation was recognized.
*It shoud be noted that particles wouldn't advance through the wall.
Since particles wouldn't advance through the wall,
"Tunneling" might be likely misleading.
Particles (or waves) continuously disappear and emerge based on the
Uncertainty Principle as
realistic intrinsic property of particles (or waves).
(Particles occasionally emerge outside of the wall.)
*In addition, it shoud be noted that "Uncertainty" means "Non-Causality."
"Causality" couldn't be expected where "Uncertainty" dominates.
"Quantum Tunnelling in Wikipedia"
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