Erasmus : For the Sake of familiarity, I will present the Schrödinger equations, although the mathematical veracity of the truth of them is far beyond our current discussion.
Spherical harmonics are to the Schrödinger equation what the math of Henri Poincare was to Einstein's theory of relativity: foundational.
The foundation of the equation is structured to be a linear differential equation based on classical energy conservation, and consistent with the De Broglie relations. The solution is the wave function ψ, which contains all the information that can be known about the system.
In the Copenhagen interpretation, the modulus of ψ is related to the probability the particles are in some spatial configuration at some instant of time. Solving the equation for ψ can be used to predict how the particles will behave under the influence of the specified potential and with each other.
In words, the equation states:
When the Hamiltonian operator acts on a certain wave function Ψ, and the result is proportional to the same wave function Ψ, then Ψ is a stationary state, and the proportionality constant, E, is the energy of the state Ψ.
The time-independent Schrödinger equation is discussed further below. In linear algebra terminology, this equation is an eigenvalue equation.
As before, the most famous manifestation is the non-relativistic Schrödinger equation for a single particle moving in an electric field (but not a magnetic field):
Where μ is the particle's "reduced mass", V is its potential energy, ∇2 is the Laplacian (a differential operator), and Ψ is the wave function (more precisely, in this context, it is called the "position-space wave function").
In plain language, it means "total energy equals kinetic energy plus potential energy", but the terms take unfamiliar forms for reasons explained below.
A wave function that satisfies the non-relativistic Schrödinger equation with V = 0. In other words, this corresponds to a particle traveling freely through empty space. The real part of the wave function is plotted here. where i is the imaginary unit, ħ is the Planck constant divided by 2π, the symbol ∂/∂t indicates a partial derivative with respect to time t, Ψ (the Greek letter Psi) is the wave function of the quantum system, and Ĥ is the Hamiltonian operator (which characterizes the total energy of any given wave function and takes different forms depending on the situation).
The time-independent Schrödinger equation predicts that wave functions can form standing waves, called stationary states (also called "orbitals", as in atomic orbitals or molecular orbitals).
Time dependent Schrondinger Equation
Time INdependent Schrondinger Equation
The term "Schrödinger equation" can refer to both the general equation (first box above), or the specific nonrelativistic version (second box above and variations thereof). The general equation is indeed quite general, used throughout quantum mechanics, for everything from the Dirac equation to quantum field theory, by plugging in various complicated expressions for the Hamiltonian.
The specific nonrelativistic version is a simplified approximation to reality, which is quite accurate in many situations, but very inaccurate in others (see relativistic quantum mechanics and relativistic quantum field theory).
To apply the Schrödinger equation, the Hamiltonian operator is set up for the system, accounting for the kinetic and potential energy of the particles constituting the system, then inserted into the Schrödinger equation. The resulting partial differential equation is solved for the wave function, which contains information about the system.
The Schrödinger equation provides a way to calculate the wave function of a system and how it changes dynamically in time. However, the Schrödinger equation does not directly say what, exactly, the wave function is.
Interpretations of quantum mechanics address questions such as what the relation is between the wave function, the underlying reality, and the results of experimental measurements.
Each of these three rows is a wave function which satisfies the time-dependent Schrödinger equation for a harmonic oscillator.
Left: The real part (blue) and imaginary part (red) of the wave function. Right: The probability of finding the particle with this wave function at a given position.
The top two rows are examples of stationary states, which correspond to standing waves. The bottom row is an example of a state which is not a stationary state. The right column illustrates why stationary states are called "stationary".
Erasmus : Time-dependent equation
The form of the Schrödinger equation depends on the physical situation (see below for special cases). The most general form is the time-dependent Schrödinger equation, which gives a description of a system evolving with time:
Erasmus : The electron is a subatomic particle, symbol e− or β−, with a negative elementary electric charge. Electrons belong to the first generation of the lepton particle family
Composition |
Elementary particle |
Statistics |
Fermionic |
Generation |
First |
Interactions |
Gravity, electromagnetic, weak |
Symbol |
e−, β− |
Antiparticle |
Positron (also called antielectron) |
Discovered |
J. J. Thomson (1897 |
Mass |
9.10938291(40)×10−31 kg |
Electric charge |
−1 e |
Magnetic moment |
−1.00115965218076(27) uB |
Spin |
1⁄2 |
Dr AXxxxx : We have come a long way. Of Course! It is Concerning to those of us, charged to see.
Big Bang and Matter
Nature of Matter in Universe
Dark Matter
NonBaryonic Matter
Non Baryonic Matter
Baryons
Baryonic Matter
Erasmus :The gravitational force is extremely weak compared with other fundamental forces. For example, the gravitational force between an electron and proton one meter apart is approximately 10×10−67 N, whereas the electromagnetic force between the same two particles is approximately 10×10−28 N.
Both these forces are weak when compared with the forces we are able to experience directly, but the electromagnetic force in this example is some 39 orders of magnitude (i.e. 1039) greater than the force of gravity—roughly the same ratio as the mass of the Sun compared to a microgram.
Erasmus :Neutrons and protons are both nucleons, which are attracted and bound together by the nuclear force to form atomic nuclei. The nucleus of the most common isotope of the hydrogen atom (with the chemical symbol "H") is a lone proton.
The nuclei of the heavy hydrogen isotopes deuterium and tritium contain one proton bound to one and two neutrons, respectively. All other types of atomic nuclei are composed of two or more protons and various numbers of neutrons. The most common nuclide of the common chemical element lead, 208Pb has 82 protons and 126 neutrons, for example.
Erasmus : The free neutron has a mass of about 1.675×10−27 kg (equivalent to 939.6 MeV/c2, or 1.0087u). The neutron has a mean square radius of about 0.8×10−15 m, or 0.8 fm and it is a spin-½ fermion. The neutron has a magnetic moment with a negative value, because its orientation is opposite to the neutron's spin.
The neutron's magnetic moment causes its motion to be influenced by magnetic fields. Although the neutron has no net electric charge, it does have a slight distribution of charge within it. With its positive electric charge, the proton is directly influenced by electric fields, whereas the response of the neutron to this force is much weaker.
Erasmus : A free neutron is unstable, decaying to a proton, electron and antineutrino with a mean lifetime of just under 15 minutes (881.5±1.5 s). This radioactive decay, known as beta decay, is possible since the mass of the neutron is slightly greater than the proton.
The free proton is stable. Neutrons or protons bound in a nucleus can be stable or unstable, however, depending on the nuclide. Beta decay, in which neutrons decay to protons, or vice versa, is governed by the weak force, and it requires the emission or absorption of electrons and neutrinos, or their antiparticles.
In physics, the proton-to-electron mass ratio, μ or β, is simply the rest mass of the proton divided by that of the electron. Because this is a ratio of like-dimensioned physical quantity, it is a dimensionless quantity, a function of the dimensionless physical constants, and has numerical value independent of the system of units, namely:
μ = mp/me = 1836.15267245(75) .
The number enclosed in parentheses is the measurement uncertainty on the last two digits. The value of μ is known to about 0.4 parts per billion.
μ is a fundamental physical constant because:
- Nearly all of science deals with baryonic matter and how the fundamental interactions affect such matter. Baryonic matter consists of quarks and particles made from quarks, like protons and neutrons.
Free neutrons have a half-life of 613.9 seconds. Electrons and protons appear to be stable, to the best of current knowledge. (Theories of proton decay predict that the proton has a half-life on the order of at least 1032 years. To date, there is no experimental evidence of proton decay.);
- The proton is the most important baryon, while the electron is the most important lepton;
Kinkajou :
Discoveries in Physics