Hamiltonian (quantum mechanics)

Last updated

In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's energy spectrum or its set of energy eigenvalues, is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

Contents

The Hamiltonian is named after William Rowan Hamilton, who developed a revolutionary reformulation of Newtonian mechanics, known as Hamiltonian mechanics, which was historically important to the development of quantum physics. Similar to vector notation, it is typically denoted by , where the hat indicates that it is an operator. It can also be written as or .

Introduction

The Hamiltonian of a system is the sum of the kinetic energies of all the particles, plus the potential energy of the 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.

Schrödinger Hamiltonian

One particle

By analogy with classical mechanics, the Hamiltonian is commonly expressed as the sum of operators corresponding to the kinetic and potential energies of a system in the form

where

is the potential energy operator and

is the kinetic energy operator in which is the mass of the particle, the dot denotes the dot product of vectors, and

is the momentum operator where a is the del operator. The dot product of with itself is the Laplacian . In three dimensions using Cartesian coordinates the Laplace operator is

Although this is not the technical definition of the Hamiltonian in classical mechanics, it is the form it most commonly takes. Combining these yields the familiar form used in the Schrödinger equation:

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.

Many particles

The formalism can be extended to particles:

where

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, [1] 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.

Schrödinger equation

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

is a unitary operator. It is the time evolution operator, or propagator , of a closed quantum system. If the Hamiltonian is time-independent, form a one parameter unitary group (more than a semigroup); this gives rise to the physical principle of detailed balance.

Dirac formalism

However, in the more general formalism of Dirac, the Hamiltonian is typically implemented as an operator on a Hilbert space in the following way:

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:

Since 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 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. [2] 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

Free 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:

Constant-potential well

For a particle in a region of constant potential (no dependence on space or time), in one dimension, the Hamiltonian is:

in three dimensions

This applies to the elementary "particle in a box" problem, and step potentials.

Simple harmonic oscillator

For a simple harmonic oscillator in one dimension, the potential varies with position (but not time), according to:

where the angular frequency , effective spring constant , and mass of the oscillator satisfy:

so the Hamiltonian is:

For three dimensions, this becomes

where the three-dimensional position vector using cartesian coordinates is (, , ), its magnitude is

Writing the Hamiltonian out in full shows it is simply the sum of the one-dimensional Hamiltonians in each direction:

Rigid rotor

For a rigid rotor—i.e., system of particles which can rotate freely about any axes, not bound in any potential (such as free molecules with negligible vibrational degrees of freedom, say due to double or triple chemical bonds), the Hamiltonian is:

where , , and are the moment of inertia components (technically the diagonal elements of the moment of inertia tensor), and , and are the total angular momentum operators (components), about the , , and axes respectively.

Electrostatic or coulomb potential

The Coulomb potential energy for two point charges and (i.e., those that have no spatial extent independently), in three dimensions, is (in SI units—rather than Gaussian units which are frequently used in electromagnetism):

However, this is only the potential for one point charge due to another. If there are many charged particles, each charge has a potential energy due to every other point charge (except itself). For charges, the potential energy of charge due to all other charges is (see also Electrostatic potential energy stored in a configuration of discrete point charges): [3]

where is the electrostatic potential of charge at . The total potential of the system is then the sum over :

so the Hamiltonian is:

Electric dipole in an electric field

For an electric dipole moment constituting charges of magnitude , in a uniform, electrostatic field (time-independent) , positioned in one place, the potential is:

the dipole moment itself is the operator

Since the particle is stationary, there is no translational kinetic energy of the dipole, so the Hamiltonian of the dipole is just the potential energy:

Magnetic dipole in a magnetic field

For a magnetic dipole moment in a uniform, magnetostatic field (time-independent) , positioned in one place, the potential is:

Since the particle is stationary, there is no translational kinetic energy of the dipole, so the Hamiltonian of the dipole is just the potential energy:

For a spin-½ particle, the corresponding spin magnetic moment is: [4]

where is the spin gyromagnetic ratio (a.k.a. "spin g-factor"), is the electron charge, is the spin operator vector, whose components are the Pauli matrices, hence

Charged particle in an electromagnetic field

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. [1] The canonical momentum operator , which includes a contribution from the field and fulfils the canonical commutation relation, must be quantized;

,

where is the kinetic momentum operator. The quantization prescription reads

,

so the corresponding kinetic energy operator is

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 operator commutes 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 :

Therefore,

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

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:

where

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

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

Similarly, one can show that

If we define "conjugate momentum" variables by

then the above equations become

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

See also

Related Research Articles

Quantum harmonic oscillator Important, well-understood quantum mechanical model

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.

Schrödinger equation Linear partial differential equation whose solution describes the quantum-mechanical system.

The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the best known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the wave functions of atomic nuclei and electrons in a molecule can be treated separately, based on the fact that the nuclei are much heavier than the electrons. The approach is named after Max Born and J. Robert Oppenheimer who proposed it in 1927, in the early period of quantum mechanics.

In physics, an operator is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry. Because of this, they are very useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.

Blochs theorem Fundamental theorem in condensed matter physics

In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. Mathematically, they are written:

Path integral formulation Formulation of quantum mechanics

The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.

Creation and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator lowers the number of particles in a given state by one. A creation operator increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator. In many subfields of physics and chemistry, the use of these operators instead of wavefunctions is known as second quantization.

In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities. For example,

Canonical quantization Process of converting a classical physical theory into one compatible with quantum mechanics

In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory, to the greatest extent possible.

In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. In quantum mechanics, it means the particle is in a region of uniform potential, usually set to zero in the region of interest since the potential can be arbitrarily set to zero at any point in space.

In quantum mechanics, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter, to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics.

In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimension, the definition is:

The Gross–Pitaevskii equation describes the ground state of a quantum system of identical bosons using the Hartree–Fock approximation and the pseudopotential interaction model.

In quantum mechanics, the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-½ particles, which takes into account the interaction of the particle's spin with an external electromagnetic field. It is the non-relativistic limit of the Dirac equation and can be used where particles are moving at speeds much less than the speed of light, so that relativistic effects can be neglected. It was formulated by Wolfgang Pauli in 1927.

In quantum mechanics and quantum field theory, a Schrödinger field, named after Erwin Schrödinger, is a quantum field which obeys the Schrödinger equation. While any situation described by a Schrödinger field can also be described by a many-body Schrödinger equation for identical particles, the field theory is more suitable for situations where the particle number changes.

This article relates the Schrödinger equation with the path integral formulation of quantum mechanics using a simple nonrelativistic one-dimensional single-particle Hamiltonian composed of kinetic and potential energy.

This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.

In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator.

In quantum mechanics, a translation operator is defined as an operator which shifts particles and fields by a certain amount in a certain direction.

The Peierls substitution method, named after the original work by Rudolf Peierls is a widely employed approximation for describing tightly-bound electrons in the presence of a slowly varying magnetic vector potential.

References

  1. 1 2 Resnick, R.; Eisberg, R. (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd ed.). John Wiley & Sons. ISBN   0-471-87373-X.
  2. Atkins, P. W. (1974). Quanta: A Handbook of Concepts. Oxford University Press. ISBN   0-19-855493-1.
  3. Grant, I. S.; Phillips, W. R. (2008). Electromagnetism . Manchester Physics Series (2nd ed.). ISBN   978-0-471-92712-9.
  4. Bransden, B. H.; Joachain, C. J. (1983). Physics of Atoms and Molecules. Longman. ISBN   0-582-44401-2.