The quantum Hamiltonian is the physical state of a system, which 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). (See Mathematical formulation of quantum mechanics) In the physical sciences, a phase is a set of states of a macroscopic physical system that have relatively uniform chemical composition and physical properties (i. ... System (from the Latin (systēma), and this from the Greek (sustēma)) is an assemblage of entity/objects, real or abstract, comprising a whole with each and every component/element interacting or related to another one. ... Ray may refer to: A ray, or half-line in geometry or physics. ... In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ... A density matrix, or density operator, is used in quantum theory to describe the statistical state of a quantum system. ... In mathematics, a bounded linear operator A over a Hilbert space H is said to be in the trace class if for some (and hence all) orthonormal bases {ek}k of H the sum of positive terms is finite. ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... One of the remarkable characteristics of the mathematical formulation of quantum mechanics, which distinguishes it from mathematical formulations of theories developed prior to the early 1900s, is its use of abstract mathematical structures, such as Hilbert spaces and operators on these spaces. ...

Physically observable quantities are described by self-adjoint operators 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 L²(R), the space of complex-valued and square-integrable (with respect to the Lebesgue measure) functions on the real line. On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ... In mathematics, the Lp and lp spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ...

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 mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ...

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

The eigenkets (eigenvectors) of H, denoted

(using Dirac 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_a}, solving the equation:

.

Since H is a Hermitian operator, the energy is always a real number.

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 (in the infinite-dimensional case, the set of eigenvalues need not coincide with the spectrum). However, all routine quantum mechanical calculations can be done using the physical formulation. In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ... In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ... In functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of eigenvalues, which is much more useful in the case of operators on infinite-dimensional spaces. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ... In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ... In most modern usages of the word spectrum, there is a unifying theme of between extremes at either end. ...

As with all observables, the spectrum of the Hamiltonian are the possible outcomes when one measures the total energy of a system. 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. (See here for details.) The pure point spectrum can be associated to eigenvectors, which in turn are the bound states of the system. The absolutely continuous spectrum correspond to the free states. The singular spectrum, interestingly enough, are 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. 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 operations. ... In most modern usages of the word spectrum, there is a unifying theme of between extremes at either end. ... In functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of eigenvalues, which is much more useful in the case of operators on infinite-dimensional spaces. ... In mathematics, projection-valued measures are used to express results in spectral theory. ... In functional analysis, the spectrum of an operator generalizes the notion of eigenvalues. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In physics, a bound state is a composite of two or more building blocks (particles or bodies) that behaves as a single object. ... The free states of the United States existed in opposition to the slave states prior to the American Civil War. ...

The Hamiltonian generates the time evolution of quantum states. If is the state of the system at time t, then A pocket watch, a device used to measure time. ...

.

where is h-bar. This equation is known as the Schrödinger equation. (It takes the same form as the Hamilton-Jacobi equation, which is one of the reasons H is also called the Hamiltonian.) 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 A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, is the definition of energy of a quantum system. ... The Hamilton-Jacobi-Bellman (HJB) equation is a partial differential equation which is central to optimal control theory. ...

.

In physical literature, the exponential operator on the right hand side is defined by the power series. 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 the case of the exponential function, the continuous, or just the holomorphic functional calculus suffices. We note again, however, that for common calculations the physicist's formulation is quite proficient. The exponential function is one of the most important functions in mathematics. ... In functional analysis (a branch of mathematics), a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded by the same number, over all non-zero vectors v in... In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. ... In operator theory and C*-algebra theory the continuous functional calculus allows applications of continuous functions to normal elements of a associates to a normal element C*-algebra. ... In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. ...

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

is an unitary operator. It is the time evolution operator, or propagator, of a closed quantum system. 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 functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U*U=UU*=I where I is the identity operator. ... Stones theorem on one-parameter unitary groups is a basic theorem of functional analysis which establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H and one-parameter families of unitary operators which are strongly continuous, that is and are homomorphisms: Such one-parameter... In mathematics, a C0-semigroup is a continuous morphism from (R+,+) into a topological monoid, usually L(H), the algebra of linear continuous operators on some Hilbert space H. Thus, strictly speaking, not the C0-semigroup, but rather its image, is a semigroup. ... In mathematics, and in statistical mechanics in physics, a Markov process is said to show detailed balance if the transition rates between each pair of states i and j in the state space obey where P is the Markov transition matrix, ie Pij = P( Xt =i | Xt−1 = j ); and...

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. 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. The wavelength is the distance between repeating units of a wave pattern. ...

It turns out that degeneracy occurs whenever a nontrivial 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 In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse... For an electrical switch that periodically reverses the current see commutator (electric) In mathematics the commutator of two elements g and h of a group G is the element g −1 h −1 gh, often denoted by [ g, h ]. It is equal to the groups identity if and only...

Since U is nontrivial, at least one pair of and 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. This article concerns the rotation operator, as it appears in quantum mechanics. ...

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

U = I − iεG + O(ε²)

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

[H,G] = 0

Therefore,

In obtaining this result, we have used the Schrödinger equation, as well as its dual, Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ...

.

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 probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical... Gyroscope. ...

Hamilton's equations

Hamilton's equations in classical Hamiltonian mechanics have a direct analogy in quantum mechanics. Suppose we have a set of basis states , which need not necessarily be eigenstates of the energy. For simplicity, we assume that they are discrete, and that they are orthonormal, i.e., Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...

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, , can be expanded in terms of these basis states:

where

The coefficients a_n(t) are complex variables. 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. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

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

where the last step was obtained by expanding 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). With this choice of independent variables, we can calculate the partial derivative In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... 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 (as opposed to the total derivative, in which all variables are allowed to vary). ...

By applying Schrödinger's equation and using the orthonormality of the basis states, this further reduces to In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ...

Similarly, one can show that

If we define "conjugate momentum" variables π_n by

then the above equations become

which is precisely the form of Hamilton's equations, with the a_ns as the generalized coordinates, the π_ns as the conjugate momenta, and taking the place of the classical Hamiltonian.

See also

• Hamiltonian mechanics