Hamiltonian (quantum Mechanics)

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The quantum Hamiltonian, as explained in the article Mathematical Formulation Of Quantum Mechanics , the physical state of a system may be characterized as a Ray in an abstract Hilbert Space (or, in the case of Ensembles , as a Trace Class operator with trace 1). Physically observable quantities are described by Self-adjoint Operator s acting on the Hilbert space. For example, the Hilbert space associated with the spin degrees of freedom of a spin-1/2 particle is C2, while the Hilbert space associated to a spinless particle moving on a line is ''L''2('''R''') , the space of complex-valued and square-integrable (with respect to the Lebesgue Measure ) functions on the real line. The quantum Hamiltonian ''H'' is the observable corresponding to the total energy of the system. If the state space is finite dimensional, then it is of course Bounded . In the infinite dimensional case, it is almost always unbounded, therefore not defined everywhere. In introductory physics literature, the following is considered either as a definition or an axiom:
	:::<math> H \left A Ight Angle	E_a \left a ight angle</math>
	The Hamiltonian Generates The	"http://wwwinformationdelightinfo/encyclopedia/entry/time" class="copylinks">Time evolution of quantum states If <math>\left \psi (t) ight angle</math> is the state of the system at time ''t'', then
	:<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>
	It Turns Out That Degeneracy Occurs Whenever A Nontrivial	"http://wwwinformationdelightinfo/encyclopedia/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/encyclopedia/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/encyclopedia/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

:

• }}{\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 $\backslash pi\_n$s as the conjugate momenta, and $\backslash langle\; H\; angle$ taking the place of the classical Hamiltonian.


SEE ALSO

• Hamiltonian Mechanics