Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H is the observable corresponding to the total energy of the system. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Physics, particularly in Quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical As with all observables, the spectrum of the Hamiltonian is the set of possible outcomes when one measures the total energy of a system. In Functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of Eigenvalues for matrices Like any other self-adjoint operator, the spectrum of the Hamiltonian can be decomposed, via its spectral measures, into pure point, absolutely continuous, and singular parts. In Mathematics, on a finite-dimensional Inner product space, a self-adjoint operator is one that is its own adjoint, or equivalently one whose matrix In Mathematics, especially Functional analysis, the spectrum of an operator generalizes the notion of eigenvalues In Mathematics, particularly Functional analysis a projection-valued measure is a function defined on certain subsets of a fixed set and whose values are self-adjoint The pure point spectrum can be associated to eigenvectors, which in turn are the bound states of the system. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Physics, a bound state is a composite of two or more building blocks ( particles or bodies) that behaves as a single object The absolutely continuous spectrum corresponds to the free states. The singular spectrum, interestingly enough, comprises physically impossible outcomes. For example, consider the finite potential well, which admits bound states with discrete negative energies and free states with continuous positive energies.

1 Schrödinger equation
2 Energy eigenket degeneracy, symmetry, and conservation laws
3 Hamilton's equations
4 See also

Schrödinger equation

The Hamiltonian generates the time evolution of quantum states. For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of If $\left| \psi (t) \right\rangle$ is the state of the system at time t, then

$H \left| \psi (t) \right\rangle = \mathrm{i} \hbar {\partial\over\partial t} \left| \psi (t) \right\rangle$ .

where $\hbar$ is the reduced Planck constant . This equation is known as the Schrödinger equation. In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system (It takes the same form as the Hamilton-Jacobi equation, which is one of the reasons H is also called the Hamiltonian. In Physics, the Hamilton–Jacobi equation (HJE is a reformulation of Classical mechanics and thus equivalent to other formulations such as Newton's laws of ) Given the state at some initial time (t = 0), we can integrate it to obtain the state at any subsequent time. In particular, if H is independent of time, then

$\left| \psi (t) \right\rangle = \exp\left(-{\mathrm{i}Ht \over \hbar}\right) \left| \psi (0) \right\rangle$ .

Note: In introductory physics literature, the following is often taken as an assumption:

The eigenkets (eigenvectors) of H, denoted $\left| a \right\rang$ (using Dirac bra-ket notation), provide an orthonormal basis for the Hilbert space. Bra-ket notation is a standard notation for describing Quantum states in the theory of Quantum mechanics composed of angle brackets (chevrons and Vertical In Mathematics, an orthonormal basis of an Inner product space V (i This article assumes some familiarity with Analytic geometry and the concept of a limit. The spectrum of allowed energy levels of the system is given by the set of eigenvalues, denoted {E_a}, solving the equation:

$H \left| a \right\rangle = E_a \left| a \right\rangle$ . In Functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of Eigenvalues for matrices

Since H is a Hermitian operator, the energy is always a real number. In Mathematics, on a finite-dimensional Inner product space, a self-adjoint operator is one that is its own adjoint, or equivalently one whose matrix In Mathematics, the real numbers may be described informally in several different ways

From a mathematically rigorous point of view, care must be taken with the above assumption. Operators on infinite-dimensional Hilbert spaces need not have eigenvalues (the set of eigenvalues does not necessarily coincide with the spectrum of an operator). In Functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of Eigenvalues for matrices However, all routine quantum mechanical calculations can be done using the physical formulation.

Similarly, the exponential operator on the right hand side of the Schrödinger equation is sometimes defined by the power series. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) One might notice that taking polynomials of unbounded and not everywhere defined operators may not make mathematical sense, much less power series. Rigorously, to take functions of unbounded operators, a functional calculus is required. In Mathematics, specifically in Functional analysis, closed linear operators are an important class of Linear operators on Banach spaces They In Mathematics, a functional calculus is a theory allowing one to apply Mathematical functions to Mathematical operators The term was also used previously In the case of the exponential function, the continuous, or just the holomorphic functional calculus suffices. In Mathematics, the continuous functional calculus of Operator theory and C*-algebra theory allows applications of continuous functions to normal elements In Mathematics, holomorphic functional calculus is Functional calculus with Holomorphic functions That is to say given a holomorphic function &fnof We note again, however, that for common calculations the physicist's formulation is quite sufficient.

By the *-homomorphism property of the functional calculus, the operator

$U = \exp\left(-{\mathrm{i}Ht \over \hbar}\right)$

is an unitary operator. In Functional analysis, a branch of Mathematics, a unitary operator is a Bounded linear operator U H → It is the time evolution operator, or propagator, of a closed quantum system. Time evolution is the change of state brought about by the passage of Time, applicable to systems with internal state (also called stateful systems) If the Hamiltonian is time-independent, {U(t)} form a one parameter unitary group (more than a semigroup); this gives rise to the physical principle of detailed balance. In Mathematics, Stone's theorem on one-parameter Unitary groups is a basic theorem of Functional analysis which establishes a One-to-one In Mathematics, a C 0-semigroup, also known as a (strongly continuous one-parameter semigroup, is a continuous morphism from ( R ++ In Mathematics and Statistical mechanics, a Markov process is said to show detailed balance if the transition rates between each pair of states i

Energy eigenket degeneracy, symmetry, and conservation laws

In many systems, two or more energy eigenstates have the same energy. A simple example of this is a free particle, whose energy eigenstates have wavefunctions that are propagating plane waves. The energy of each of these plane waves is inversely proportional to the square of its wavelength. In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. A wave propagating in the x direction is a different state from one propagating in the y direction, but if they have the same wavelength, then their energies will be the same. When this happens, the states are said to be degenerate.

It turns out that degeneracy occurs whenever a nontrivial unitary operator U commutes with the Hamiltonian. In Mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition U^* U = UU^* In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. To see this, suppose that |a> is an energy eigenket. Then U|a> is an energy eigenket with the same eigenvalue, since

$UH |a\rangle = U E_a|a\rangle = E_a (U|a\rangle) = H \; (U|a\rangle).$

Since U is nontrivial, at least one pair of $|a\rang$ and $U|a\rang$ 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 rotation operator, which rotates the wavefunctions by some angle while otherwise preserving their shape.

The existence of a symmetry operator implies the existence of a conserved observable. In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves Let G be the Hermitian generator of U:

U = I - iε G + O (ε 2)

It is straightforward to show that if U commutes with H, then so does G:

[H, G] = 0

Therefore,

$\frac{\part}{\part t} \langle\psi(t)|G|\psi(t)\rangle= \frac{1}{\mathrm{i}\hbar} \langle\psi(t)|[G,H]|\psi(t)\rangle = 0$

In obtaining this result, we have used the Schrödinger equation, as well as its dual,

$\langle\psi (t)|H = - \mathrm{i} \hbar {\partial\over\partial t} \langle\psi(t)|$ . Bra-ket notation is a standard notation for describing Quantum states in the theory of Quantum mechanics composed of angle brackets (chevrons and Vertical

Thus, the expected value of the observable G is conserved for any state of the system. In the case of the free particle, the conserved quantity is the angular momentum. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position

Hamilton's equations

Hamilton's equations in classical Hamiltonian mechanics have a direct analogy in quantum mechanics. Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. Suppose we have a set of basis states $\left\{\left| n \right\rangle\right\}$ , which need not necessarily be eigenstates of the energy. For simplicity, we assume that they are discrete, and that they are orthonormal, i. e. ,

$\langle n' | n \rangle = \delta_{nn'}.$

Note that these basis states are assumed to be independent of time. We will assume that the Hamiltonian is also independent of time.

The instantaneous state of the system at time t, $\left| \psi\left(t\right) \right\rangle$ , can be expanded in terms of these basis states:

$|\psi (t)\rangle = \sum_{n} a_n(t) |n\rangle$

where

$a_n(t) = \langle n | \psi(t) \rangle.$

The coefficients a_n(t) are complex variables. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted We can treat them as coordinates which specify the state of the system, like the position and momentum coordinates which specify a classical system. Like classical coordinates, they are generally not constant in time, and their time dependence gives rise to the time dependence of the system as a whole.

The expectation value of the Hamiltonian of this state, which is also the mean energy, is

$\langle H(t) \rangle \ \stackrel{\mathrm{def}}{=}\ \langle\psi(t)|H|\psi(t)\rangle= \sum_{nn'} a_{n'}^* a_n \langle n'|H|n \rangle$

where the last step was obtained by expanding $\left| \psi\left(t\right) \right\rangle$ in terms of the basis states.

Each of the a_n(t)'s actually corresponds to two independent degrees of freedom, since the variable has a real part and an imaginary part. We now perform the following trick: instead of using the real and imaginary parts as the independent variables, we use a_n(t) and its complex conjugate a_n*(t). In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part. With this choice of independent variables, we can calculate the partial derivative

$\frac{\partial \langle H \rangle}{\partial a_{n'}^{*}}= \sum_{n} a_n \langle n'|H|n \rangle= \langle n'|H|\psi\rangle$

By applying Schrödinger's equation and using the orthonormality of the basis states, this further reduces to

$\frac{\partial \langle H \rangle}{\partial a_{n'}^{*}}= \mathrm{i} \hbar \frac{\partial a_{n'}}{\partial t}$

Similarly, one can show that

$\frac{\partial \langle H \rangle}{\partial a_n}= - \mathrm{i} \hbar \frac{\partial a_{n}^{*}}{\partial t}$

If we define "conjugate momentum" variables π_n by

$\pi_{n}(t) = \mathrm{i} \hbar a_n^*(t)$

then the above equations become

$\frac{\partial \langle H \rangle}{\partial \pi_{n}}= \frac{\partial a_{n}}{\partial t} \quad,\quad\frac{\partial \langle H \rangle}{\partial a_n}= - \frac{\partial \pi_{n}}{\partial t}$

which is precisely the form of Hamilton's equations, with the $a n$ s as the generalized coordinates, the $π n$ s as the conjugate momenta, and $\langle H\rangle$ taking the place of the classical Hamiltonian. In Mathematics, a partial derivative of a function of several variables is its Derivative with respect to one of those variables with the others held constant In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system

Contents

Schrödinger equation

Energy eigenket degeneracy, symmetry, and conservation laws

Hamilton's equations

See also