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Hamiltonian Quantum Mechanics
Quantum operator for the sum of energies of a system
The Hamiltonian of a system represents the total energy of the system; that is, the sum of the kinetic and potential energies of all particles associated with the system. The Hamiltonian takes different forms and can be simplified in some cases by taking into account the concrete characteristics of the system under analysis, such as single or several particles in the system, interaction between particles, kind of potential energy, time varying potential or time independent one.
which allows one to apply the Hamiltonian to systems described by a wave function. This is the approach commonly taken in introductory treatments of quantum mechanics, using the formalism of Schrödinger's wave mechanics.
One can also make substitutions to certain variables to fit specific cases, such as some involving electromagnetic fields.
The formalism can be extended to particles:
is the potential energy function, now a function of the spatial configuration of the system and time (a particular set of spatial positions at some instant of time defines a configuration) and
is the kinetic energy operator of particle , is the gradient for particle , and is the Laplacian for particle n:
Combining these yields the Schrödinger Hamiltonian for the -particle case:
However, complications can arise in the many-body problem. Since the potential energy depends on the spatial arrangement of the particles, the kinetic energy will also depend on the spatial configuration to conserve energy. The motion due to any one particle will vary due to the motion of all the other particles in the system. For this reason cross terms for kinetic energy may appear in the Hamiltonian; a mix of the gradients for two particles:
where denotes the mass of the collection of particles resulting in this extra kinetic energy. Terms of this form are known as mass polarization terms, and appear in the Hamiltonian of many electron atoms (see below).
For interacting particles, i.e. particles which interact mutually and constitute a many-body situation, the potential energy function is not simply a sum of the separate potentials (and certainly not a product, as this is dimensionally incorrect). The potential energy function can only be written as above: a function of all the spatial positions of each particle.
For non-interacting particles, i.e. particles which do not interact mutually and move independently, the potential of the system is the sum of the separate potential energy for each particle, that is
The general form of the Hamiltonian in this case is:
where the sum is taken over all particles and their corresponding potentials; the result is that the Hamiltonian of the system is the sum of the separate Hamiltonians for each particle. This is an idealized situation--in practice the particles are almost always influenced by some potential, and there are many-body interactions. One illustrative example of a two-body interaction where this form would not apply is for electrostatic potentials due to charged particles, because they interact with each other by Coulomb interaction (electrostatic force), as shown below.
The Hamiltonian generates the time evolution of quantum states. If is the state of the system at time , then
This equation is the Schrödinger equation. It takes the same form as the Hamilton-Jacobi equation, which is one of the reasons is also called the Hamiltonian. Given the state at some initial time (), we can solve it to obtain the state at any subsequent time. In particular, if is independent of time, then
The exponential operator on the right hand side of the Schrödinger equation is usually defined by the corresponding power series in . One might notice that taking polynomials or power series of unbounded operators that are not defined everywhere may not make mathematical sense. 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 physicists' formulation is quite sufficient.
By the *-homomorphism property of the functional calculus, the operator
The eigenkets (eigenvectors) of , denoted , 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 , solving the equation:
From a mathematically rigorous point of view, care must be taken with the above assumptions. Operators on infinite-dimensional Hilbert spaces need not have eigenvalues (the set of eigenvalues does not necessarily coincide with the spectrum of an operator). However, all routine quantum mechanical calculations can be done using the physical formulation.[clarification needed]
Expressions for the Hamiltonian
Following are expressions for the Hamiltonian in a number of situations. Typical ways to classify the expressions are the number of particles, number of dimensions, and the nature of the potential energy function--importantly space and time dependence. Masses are denoted by , and charges by .
General forms for one particle
The particle is not bound by any potential energy, so the potential is zero and this Hamiltonian is the simplest. For one dimension:
and in higher dimensions:
For a particle in a region of constant potential (no dependence on space or time), in one dimension, the Hamiltonian is:
For a particle with mass and charge in an electromagnetic field, described by the scalar potential and vector potential, there are two parts to the Hamiltonian to substitute for. The canonical momentum operator , which includes a contribution from the field and fulfils the canonical commutation relation, must be quantized;
and the potential energy, which is due to the field, is given by
Casting all of these into the Hamiltonian gives
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 direction is a different state from one propagating in the 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 operatorcommutes with the Hamiltonian. To see this, suppose that is an energy eigenket. Then is an energy eigenket with the same eigenvalue, since
Since is nontrivial, at least one pair of and must represent distinct states. Therefore, 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. Let be the Hermitian generator of :
It is straightforward to show that if commutes with , then so does :
In obtaining this result, we have used the Schrödinger equation, as well as its dual,
Thus, the expected value of the observable is conserved for any state of the system. In the case of the free particle, the conserved quantity is the angular momentum.
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.,
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 , , can be expanded in terms of these basis states:
The coefficients 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.
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 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 and its complex conjugate. With this choice of independent variables, we can calculate the partial derivative