Schrödinger field

Schrödinger field

In quantum mechanics and quantum field theory, a Schrödinger field 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.

A Schrödinger field is also the classical limit of a quantum Schrödinger field, a classical wave which satisfies the Schrödinger equation. Unlike the quantum mechanical wavefunction, if there are interactions between the particles the equation will be nonlinear. These nonlinear equations describe the classical wave limit of a system of interacting identical particles.

The path integral of a Schrödinger field is also known as a coherent state path integral, because the field itself is an annihilation operator whose eigenstates can be thought of as coherent states of the harmonic oscillations of the field modes.

Schrödinger fields are useful for describing Bose-Einstein condensation, superconductivity, superfluidity, and many-body theory in general. They are also a useful alternative formalism for nonrelativistic quantum mechanics.

A Schrödinger field is the nonrelativistic limit of a Klein-Gordon field.

Summary

A Schrödinger field is a quantum field whose quanta obey the Schrödinger equation. In the classical limit, it can be understood as the quantized wave equation of a Bose Einstein condensate or a superfluid.

Free Field

A Schrödinger field has the free field Lagrangian:L = psi^dagger left(i{partialover partial t} + { abla^2 over 2m} ight)psi

When psi is a complex valued field in a path integral, or equivalently an operator with canonical commutation relations, it describes a collection of identical nonrelativistic bosons. When psi is a grassmann valued field, or equivalently an operator with canonical anticommutation relations, the field describes identical fermions.

External Potential

If the particles interact with an external potential V(x), the interaction makes a local contribution to the action::S = int_{xt} psi^dagger left(i{partial over partial t} + { abla^2over 2m} ight)psi - psi^dagger(x) psi(x) V(x)

If the ordinary Schrödinger equation for V has known energy eigenstates phi_i(x) with energies E_i, then the field in the action can be rotated into a diagonal basis by a mode expansion::psi(x) = sum_i psi_i phi_i(x),The action becomes::S= int_t sum_i psi_i^daggerleft( i{partial over partial t} - E_i ight) psi_i,Which is the position-momentum path integral for a collection of independent Harmonic oscillators.

To see the equivalence, note that decomposed into real and imaginary parts the action is::S= int_t sum_i 2psi_r{dpsi_iover dt} - E_i(psi_r^2 + psi_i^2)after an integration by parts. Integrating over psi_r gives the action:S= int_t sum_i {1 over E_i} left({dpsi_iover dt} ight)^2 - E_i psi_i^2which, rescaling scriptstyle psi_i, is a harmonic oscillator action with frequency E_i.

Pair Potential

When the particles interact with a pair potential V(x_1,x_2), the interaction is a nonlocal contribution to the action::S = int_{xt} psi^dagger left(i{partial over partial t} + { abla^2 over 2m} ight)psi - int_{xy} psi^dagger(x) psi(x)V(x,y) psi^dagger(y)psi(y)

A pair-potential is the non-relativistic limit of a relativistic field coupled to electrodynamics. Ignoring the propagating degrees of freedom, the interaction between nonrelativistic electrons is the coulomb repulsion. In 3+1 dimensions, this is::V(x,y)= {q^2over |x-yWhen coupled to an external potential to model classical positions of nuclei, a Schrödinger field with this pair potential describes nearly all of condensed matter physics. The exceptions are effects like superfluidity, where the quantum mechanical interference of nuclei is important, and inner shell electrons where the electron motion can be relativistic.

Non-Linear Schrödinger Equation

A special case of a delta-function interaction V(x_1,x_2) is widely studied, and is known as the nonlinear Schrödinger equation. Because the interactions always happen when two particles occupy the same point, the action for the nonlinear Schrödinger equation is local::S = int_x psi^dagger left(i{partial over partial t} + { abla^2 over 2m} ight)psi + lambda (psi^dagger psi)^2 The interaction strength lambda requires renormalization in dimensions higher than 2 and in two dimensions it has logarithmic running. In any dimensions, and even with power-law running, the theory is well defined. If the particles are fermions, the interaction vanishes.

Many-Body Potentials

The potentials can include many-body contributions. The interacting Lagrangian is then::L_i = int_x (psi^daggerpsi)(x_1)(psi^daggerpsi)(x_2) ...(psi^daggerpsi)(x_n) V(x_1,x_2.,x_n),

These types of potentials are important in some effective descriptions of close-packed atoms. Higher order interactions are less and less important.

Canonical Formalism

The canonical momentum association with the field psi is:Pi(x) = i psi^dagger,

The canonical commutation relations are like an independent harmonic oscillator at each point:: [psi(x), psi^dagger (y)] = delta(x-y)

The field Hamiltonian is:H = S - int Pi(x) {dover dt}psi = int {| abla psi|^2 over 2m} + int_{xy} V(x,y)psi^dagger(x)psi(x)psi^dagger(y)psi(y),

and the field equation for any interaction is a nonlinear and nonlocal version of the Schrödinger equation. For pairwise interactions::i{partial over partial t} psi = -{ abla^2over 2m} psi + left(int_y V(x,y)psi^dagger(y)psi(y) ight) psi(x),

Perturbation Theory

The expansion in Feynman diagrams is called many-body perturbation theory. The propagator is

:G(x) = {1 over iomega - {k^2over 2m} },

The interaction vertex is the Fourier transform of the pair-potential. In all the interactions, the number of incoming and outgoing lines is equal.

Exposition

Identical Particles

The many body Schrödinger equation for identical particles describes the time evolution of the many-body wavefunction psi(x_1,x_2,...x_N) which is the probability amplitude for N particles to have the listed positions. The Schrödinger equation for psi is::i{dover dt} psi = ({ abla_1^2 over 2m} + { abla_2^2 over 2m} ... +{ abla_N^2 over 2m} + V(x_1,x_2,...,x_N) )psi ,With Hamiltonian:H = {p_1^2over 2m} + {p_2^2 over 2m} + ... + {p_N^2 over 2m} + V(x_1,....,x_N),Since the particles are indistinguishable, the wavefunction has some symmetry under switchingpositions. Either
#psi(x_1,x_2,...) = psi(x_2,x_1...), for bosons,
#psi(x_1,x_2,..)=-psi(x_2,x_1,...), for fermions

Since the particles are indistinguishable, the potential V must be unchanged under permutations.If:V(x_1,...,x_N) = V_1(x_1)+ V_2(x_2) ... + V_N(x_N),

then it must be the case that V_1=V_2=..=V_N . If:V(x_1 ... ,x_N) = V_{1,2}(x_1,x_2) + V_{1,3}(x_2,x_3) + V_{2,3}(x_1,x_2),

then V_{1,2} = V_{1,3} = V_{2,3} and so on.

In the Schrödinger equation formalism, the restrictions on the potential are ad-hoc, and the classical wave limit is hard to reach. It also has limited usefulness if a system is open to the environment, because particles might coherently enter and leave.

Nonrelativistic Fock Space

A Schrödinger field is defined by extending the Hilbert space of states toinclude configurations with arbitrary particle number. A nearly complete basis for this set of states is the collection::
N;x_1,ldots,x_N angle,labeled by the total number of particles and their position. An arbitrary state with particles at separated positions is described by a superposition of states of this form.:psi_0 |0 angle + int_x psi_1(x) |1;x angle + int_{x_1x_2} psi_2(x_1,x_2)|1;x_1 x_2 angle + ldots,In this formalism, keep in mind that any two states whose positions can be permuted into each other are really the same, so the integration domains need to avoid double counting. Also keep in mind that the states with more than one particle at the same point have not yet been defined. The quantity psi_0 is the amplitude that no particles are present, and its absolute square is the probability that the system is in the vacuum.

In order to reproduce the Schrödinger description, the inner product on the basis states should be:langle 1;x_1|1;y_1 angle = delta(x_1-y_1),

:langle 2;x_1 x_2 | 2;y_1 y_2 angle = delta(x_1-y_1)delta(x_2-y_2) pm delta(x_1 -y_2)delta(x_2-y_1),and so on. Since the discussion is nearly formally identical for bosons and fermions, although the physical properties are different, from here on the particles will be bosons.

There are natural operators in this Hilbert space. One operator, called scriptstyle psi^dagger(x), is the operator which introduces an extra particle at x.It is defined on each basis state::psi^dagger(x) |N;x_1 ... x_n angle = |N+1; x_1, ...,x_n, x angle,with slight ambiguity when a particle is already at x.

Another operator removes a particle at x, and is called psi. This operator is the conjugate of the operator psi^dagger. Because scriptstyle psi^dagger has no matrix elements which connect to states with no particle at x, psi must give zero when acting on such a state.:psi(x) |N; x_1 ... ,x_N angle = delta(x-x_1) |N-1;x_2 ...,x_N angle + delta(x-x_2)|N-1;x_1,x_3...,x_ angle + ldots ,

The position basis is an inconvenient way to understand coincident particles because states with a particle localized at one point have infinite energy, so intuition is difficult. In order to see what happens when two particles are at exactly the same point, it is mathematically simplest either to make space into a discrete lattice, or to Fourier transform the field in a finite volume.

The operator:psi^dagger(k)= int_x e^{-ikx} psi^dagger(x),creates a superposition of one particle states in a plane wave state with momentum k, in other words, it produces a new particle with momentum k. The operator:psi(k) = int_x e^{ikx} psi(x),annihilates a particle with momentum k.

If the potential energy for interaction of infinitely distant particles vanishes, the fourier transformed operators in infinite volume create states which are noninteracting. The states are infinitely spread out, and the chance that the particles are nearby is zero.

The matrix elements for the operators between non-coincident points reconstructs the matrix elements of the Fourier transform between all modes:
#psi^dagger(k) psi^dagger(k') - psi^dagger(k')psi^dagger(k) =0,
#psi(k)psi(k') - psi(k')psi(k) =0,
#psi(k)psi^dagger(k') - psi(k')psi^dagger(k) = delta(k-k'),where the delta function is either the Dirac delta function or the Kronecker delta, depending on whether the volume is infinite or finite.

The commutation relations now determine the operators completely, and when the spatial volume is finite, there are no conceptual hurdle to understand coinciding momenta because momenta are discrete. In a discrete momentum basis, the basis states are::
n_1, n_2, ... n_k angle,where the n's are the number of particles at each momentum. For fermions and anyons, the number of particles at any momentum is always either zero or one. The operators scriptstyle psi_k have harmonic-oscillator like matrix elements between states, independent of the interaction::psi^dagger(k)|..,n_k,ldots angle = sqrt{n_k+1}, |...,n_k+1,ldots angle

:psi(k) | ...,n_k, ldots angle = sqrt{n_k}, |...,n_k-1,ldots angle

So that the operator:sum_k psi^dagger(k)psi(k) = int_x psi^dagger(x)psi(x)counts the total number of particles.

Now it is easy to see that the matrix elements of scriptstyle psi(x) and scriptstyle psi^dagger(x) have harmonic oscillator commutation relations too.
# [psi(x),psi(y)] = [psi^dagger(x),psi^dagger(y)] = 0
# [psi(x),psi^dagger(y)] = delta(x-y) So that there really is no difficulty with coincident particles in position space.

The operator scriptstyle psi^dagger(x) psi(x) which removes and replaces a particle, acts as a sensor to detect if a particle is present at x. The operator scriptstyle psi^dagger ablapsi acts to multiply the state by the gradient of the many body wavefunction. The operator:H= int_x psi^dagger(x) { abla^2 over 2m } psi(x),acts to reproduce the right hand side of the Schrödinger equation when acting on any basis state, so that:psi^dagger i{dover dt} psi = psi^dagger {- abla^2 over 2m} psi,holds as an operator equation. Since this is true for an arbitrary state, it is also true without the scriptstyle psi^dagger.:i {partial over partial t} psi = {- abla^2 over 2m} psi,

To add interactions, add nonlinear terms in the field equations. The field form automatically ensures that the potentials obey the restrictions from symmetry.

Field Hamiltonian

The field Hamiltonian which reproduces the equations of motion is:H= { abla psi^dagger ablapsi over 2m}

The Heisenberg equations of motion for this operator reproduces the equation of motion for the field.

To find the classical field Lagrangian, apply a Legendre transform to the classical limit of the Hamiltonian.:L = psi^dagger (i {partial over partial t} + { abla^2 over 2m} )psi,

Although this is correct classically, the quantum mechanical transformation is not completely conceptually straightforward because the path integral is over eigenvalues of operators psi which are not hermitian and whose eigenvalues are not orthogonal. The path integral over field states therefore seems naively to be overcounting. This is not the case, because the time derivative term in L includes the overlap between the different field states.

Further reading

ee also

*Schrodinger equation

External links


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