Hamiltonian (quantum Mechanics)

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Hamiltonian (quantum Mechanics)


	:<math> H \left \psi (t) Ight Angle	\mathrm{i} \hbar {\partial\over\partial t} \left \psi (t) ight angle</math>
	:<math> \left \psi (t) Ight Angle	\exp\left(-{\mathrm{i}Ht \over \hbar} ight) \left \psi (0) ight angle</math>
	:The Eigenkets (eigenvectors) Of ''H'', Denoted <math>\left A Ight Ang</math> (using Dirac	"http://wwwinformationdelightinfo/information/entry/bra-ket_notation" class="copylinks">Bra-ket Notation ), provide an Orthonormal Basis for the Hilbert space The Spectrum of allowed energy levels of the system is given by the set of eigenvalues, denoted {''E''<sub>a</sub>}, solving the equation:
	::<math> H \left A Ight Angle	E_a \left a ight angle</math>
	It Turns Out That Degeneracy Occurs Whenever A Nontrivial	"http://wwwinformationdelightinfo/information/entry/Unitary_matrix" class="copylinks">Unitary Operator ''U'' Commutes with the Hamiltonian To see this, suppose that a> is an energy eigenket Then ''U''a> is an energy eigenket with the same eigenvalue, since
	:<math>UH a Angle	U E_aa angle = E_a (Ua angle) = H \ (Ua angle) </math>
	Since ''U'' Is Nontrivial, At Least One Pair Of <math>a Ang</math> And <math>Ua Ang</math> Must Represent Distinct States Therefore, ''H'' Has At Least One Pair Of Degenerate Energy Eigenkets In The Case Of The Free Particle, The Unitary Operator Which Produces The Symmetry Is The	"http://wwwinformationdelightinfo/information/entry/rotation_operator" class="copylinks">Rotation Operator , which rotates the wavefunctions by some angle while otherwise preserving their shape
		rac{1}{\mathrm{i}\hbar} \langle\psi(t) {Link without Title} \psi(t) angle
	:<math> \langle\psi (t)H	- \mathrm{i} \hbar {\partial\over\partial t} \langle\psi(t)</math>
	Hamilton's Equations In Classical	"http://wwwinformationdelightinfo/information/entry/Hamiltonian_mechanics" class="copylinks">Hamiltonian Mechanics have a direct analogy in quantum mechanics Suppose we have a set of basis states <math>\left\{\left n ight angle ight\}</math>, which need not necessarily be eigenstates of the energy For simplicity, we assume that they are discrete, and that they are orthonormal, ie,
	:<math> \langle N' N Angle	\delta_{nn'} </math>
	:<math> \psi (t) Angle	\sum_{n} a_n(t) n angle </math>
	:<math> A N(t)	\langle n \psi(t) angle </math>

\sum_{nn'} a_{n'}^ a_n \langle n'Hn angle </math>

\langle n'H\psi angle

= \mathrm{i} \hbar rac{\partial a_{n'}}{\partial t}

Similarly, one can show that

:

rac{\partial \langle H 
angle}{\partial a_n}

}}{\partial t}

If we define "conjugate momentum" variables ''π_n'' by

(t)

then the above equations become

:

rac{\partial \langle H 
angle}{\partial \pi_{n}}
= rac{\partial a_{n}}{\partial t} \quad,\quad
rac{\partial \langle H 
angle}{\partial a_n}
= - rac{\partial \pi_{n}}{\partial t}

which is precisely the form of Hamilton's equations, with the

a_n

s as the generalized coordinates, the

\pi_n

s as the conjugate momenta, and

\langle H
angle

taking the place of the classical Hamiltonian.

SEE ALSO

Hamiltonian Mechanics