Wave Mechanics

In the Mathematical Formulation Of Quantum Mechanics , each system is associated with a Complex Hilbert Space such that each instantaneous state of the system is described by a Unit Vector in that space. This state vector encodes the probabilities for the outcomes of all possible measurements applied to the system. 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 (t) Ight Angle	i \hbar {\partial\over\partial t} \left \psi (t) ight angle</math>
	:<math> H \psi N(x) Ang	E_n \psi_n(x) ang </math>
	:<math> I \hbar {\partial\over\partial T} \left \psi N (t) Ight Angle	E_n \psi_n(t) ang </math>
	:<math> \left \psi (t) Ight Angle	e^{-i E t / \hbar} \psi(0) ang </math>
	Energy Eigenstates Are Convenient To Work With Because Their Time-dependence Is So Simple That Is Why The Time-independent Schr&oumldinger Equation Is So Useful We Can Always Choose A Set Of ''instantaneous'' Energy Eigenstates Whose State Vectors <nowiki>{</nowiki>''n''><nowiki>}</nowiki> Form A	"http://wwwinformationdelightinfo/encyclopedia/entry/Vrhbosna/basis_(linear_algebra)" class="copylinks">Basis for the state space Then any state vector ''&psi(t)''> can be written as a Linear Superposition of energy eigenstates:
	:<math> \int \ \psi(\mathbf{r}, T)^2 \ D^3r	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.:

::

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

where ''P''(''x'', ''t'') 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 ,

:<math> J(x,t) A^2 {k \hbar \over m}</math>

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Wave Mechanics