Wave Model

In Physics , the Schrödinger equation, proposed by the Austrian Physicist Erwin Schrödinger in 1926 , describes the space- and Time-dependence of Quantum Mechanical systems. It is of central importance in non-relativistic quantum mechanics, playing a role for microscopic particles analogous to Newton's Second Law in Classical Mechanics for macroscopic particles. Microscopic particles include Elementary Particles , such as Electrons , as well as systems of particles, such as Atomic Nuclei . Macroscopic particles vary in mass from minute dust particles to the heaviest planets. HISTORICAL BACKGROUND AND DEVELOPMENT Schrödinger's equation follows very naturally from earlier developments: In 1905, by considering the Photoelectric Effect , Albert Einstein had published his :: $E = h f\;$ formula for the relation between the Energy E and Frequency '''f''' of the quanta of radiation (photons), where '''h''' is Planck's Constant . In 1924 Louis De Broglie presented his De Broglie Hypothesis which states that ''all'' particles (not just photons) have an associated wavefunction $\psi\;$ with properties: :: $p=h / \lambda\;$ , where $\lambda\,$ is the Wavelength of the wave and p the Momentum of the particle. De Broglie showed that this was consistent with Einstein's formula and Special Relativity so that :: $E = h f\;$ still holds, but now this is hypothesized to hold for ''all'' particles, not just photons anymore. Expressed in terms of Angular Frequency $\omega = 2\pi f\;$ and Wavenumber $k = 2\pi / \lambda\;$ , with $\hbar = h / 2 \pi\;$ we get: :: $E=\hbar \omega$ and :: $\underline{p}=\hbar \underline{k}\;$ where we have expressed p and k as Vector s. Schrödinger's great insight, late in 1925, was to express the Phase of a Plane Wave as a Complex Phase Factor : :: $\psi \approx \exp(i(\underline{k}.\underline{x}- \omega t))$ and to realize that since :: $rac{\partial}{\partial t} \psi = -i\omega \psi$ then :: $E \psi = \hbar \omega \psi = i\hbarrac{\partial}{\partial t} \psi$ and similarly since: :: $rac{\partial}{\partial x} \psi = i k_x \psi$ then :: $p_x \psi = \hbar k_x \psi = -i\hbarrac{\partial}{\partial x} \psi$ and hence :: $p_x^2 \psi = -\hbar^2rac{\partial^2}{\partial x^2} \psi$ so that :: $p^2 \psi = (p_x^2 + p_y^2 + p_z^2) \psi = -\hbar^2(rac{\partial^2}{\partial x^2} + rac{\partial^2}{\partial y^2} + rac{\partial^2}{\partial z^2}) \psi = -\hbar^2 abla^2 \psi$ And by inserting these expressions into the Newtonian Formula for a particle with total energy E, Mass '''m''', moving in a Potential '''V''': :: $E=rac{p^2}{2m}+V$ (simply the sum of the Kinetic Energy and Potential Energy ) he got his famed equation for a single particle in the 3-dimensional case in the presence of a potential: :: $i\hbarrac{\partial}{\partial t}\psi=-rac{\hbar^2}{2m} abla^2\psi + V\psi$ Using this equation, Schrödinger computed the Spectral Line s for hydrogen by treating a Hydrogen atom's single negatively Charged Electron as a wave, $\psi\;$ , moving in a Potential Well , V, created by the positively charged Proton . This tallied with experiment, the Bohr Model and also the results of Werner Heisenberg 's Matrix Mechanics - but without having to introduce Heisenberg's concept of Non-commuting Observable s. Schrödinger published his wave equation and the spectral analysis of hydrogen in a series of four papers in 1926. The Schrödinger equation defines the behaviour of $\psi\;$ , but does not interpret what $\psi\;$ ''is''. Schrödinger tried unsuccessfully to interpret it as a charge density. In 1926 Max Born , just a few days after Schrödinger fourth and final paper was published, successfully interpreted it as a Probability Amplitude , although Schrödinger was never reconciled to this Statistical or probabilistic approach. MATHEMATICAL FORMULATION In the Mathematical Formulation Of Quantum Mechanics , a physical system is associated with a Complex Hilbert Space such that each instantaneous state of the system is described by a ray (a one-dimensional subspace) in that space. The nonzero elements of a Hilbert space are by definition normalizable and it is convenient, although not necessary, to represent a state by an element of the ray which is normalized to unity. This vector is often somewhat loosely referred to as Wave Function , although in a more rigorous formulation of quantum mechanics a wave function is a special case of a state vector. (In fact, a wave function is a state in the position representation, see below). A state vector encodes the probabilities for the outcomes of all possible measurements applied to the system. It contains all information of the system that is knowable in a quantum mechanical sense. As the state of a system generally changes over time, the state vector is a function of time. The Schrödinger equation provides a quantitative description of the rate of change of the state vector.
	:<math>H(t)\left\psi\left(t Ight) Ight Angle	\mathrm{i}\hbar rac{d}{d t} \left \psi \left(t ight) ight angle</math>
	For Every Time-independent Hamiltonian, <math>H</math>, There Exists A Set Of Quantum States, <math>\left\psi N Ight Ang</math>, Known As '''energy Eigenstates''', And Corresponding Real Numbers <math>E N</math> Satisfying The	"http://wwwinformationdelightinfo/information/entry/Eigenvalue,_eigenvector_and_eigenspace" class="copylinks">Eigenvalue Equation ,
	::<math> H \left\psi N Ight Ang	E_n \left\psi_n ight ang </math>
	:<math>\mathrm{i} \hbar Rac{\partial}{\partial T} \left \psi N \left(t Ight) Ight Angle	E_n \left\psi_n\left(t ight) ight ang </math>
	:<math> \left \psi \left(t Ight) Ight Angle	\mathrm{e}^{-\mathrm{i} Et / \hbar} \left\psi\left(0 ight) ight ang </math>
	\psi(0)^\psi(0)	\psi(0)^2,
	Energy Eigenstates Are Convenient To Work With Because They Form A Complete Set Of States That Is, The Eigenvectors <math> \left\{\leftn Ight Ang Ight\} </math> Form A	"http://wwwinformationdelightinfo/information/entry/basis_(linear_algebra)" class="copylinks">Basis for the state space We introduced here the short-hand notation
	<math>\,n\, Ang	\psi_n</math>
	<math> \left\psi\left(t Ight) Ight Ang </math> Can Be Written As A	"http://wwwinformationdelightinfo/information/entry/linear_combination" class="copylinks">Linear Superposition of energy eigenstates:
	:<math>\left\psi\left(t Ight) Ight Ang	\sum_n c_n(t) \leftn ight ang \quad,\quad H \leftn ight ang = E_n \leftn ight ang \quad,\quad \sum_n \leftc_n\left(t ight) ight^2 = 1</math>
	Therefore, If We Know The Decomposition Of <math> \left\psi\left(t Ight) Ight Ang </math> Into The Energy Basis At Time <math>t	0</math>, its value at any subsequent time is given simply by
	:<math>\left\psi\left(t Ight) Ight Ang	\sum_n \mathrm{e}^{-\mathrm{i}E_nt/\hbar} c_n\left(0 ight) \leftn ight ang </math>
	H\,\,1\, Angle	E \,1\, angle \quad \hbox{and} \quad H\,\,2\, angle = E \,2\, angle
	\mathrm{e}^{-\mathrm{i}Et/\hbar} C 2 \,2\, Angle	\mathrm{e}^{-\mathrm{i}Et/\hbar}
	\left( C 1 \,1\, Angle + C 2 \,2\, Angle Ight)	\mathrm{e}^{-\mathrm{i}Et/\hbar}\,\psi(0)\, angle,
		E \mathrm{e}^{-\mathrm{i}Et/\hbar}\,\psi(0)\, angle = E\,\,\psi(t)\, angle
	:<math>\int \left\mathbf{r} Ight Angle \left\langle \mathbf{r} Ight \mathrm{d}^3 \mathbf{r}	\mathbf{I}</math>
	:<math> \int \ \left\psi\left(\mathbf{r}, T Ight) Ight^2 \ \mathrm{d}^3\mathbf{r}	1 </math>

abla \psi - \psi

} ight) = {\hbar \over m} \operatorname{Im} \left( \psi ^{---}

and measured in units of (probability)/(area × time) = ''r''⁻²''t''⁻¹.

The probability flux satisfies a quantum Continuity Equation , i.e.:

::

{ \partial \over \partial t} P\left(x,t
ight) + 
abla \cdot \mathbf{j} = 0

where

P\left(x, t
ight)

is the Probability Density and measured in units of (probability)/(volume) = ''r''⁻³.
This equation is the mathematical equivalent of Probability Conservation Law .

It is easy to show that for a Plane Wave ,

:

\psi (x,t) = A e^{ \mathrm{i} k x} e^{ - \mathrm{i} \omega t}

the probability flux is given by

	CATEGORIES ABOUT SCHRöDINGER EQUATION

	fundamental physics concepts

	schrodinger equation

	partial differential equations

	quantum mechanics

	equations

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Information About

Wave Model