Annihilation and creation operators in quantum mechanics are constructed in the framework of Nelson's stochastic mechanics. Employing Ito's stochastic calculus, annihilation and creation operators were found to be written as (1/\ensuremath{\surd}2\ensuremath{\omega})[d/dt,\ensuremath{\partial}/\ensuremath{\partial}B\ifmmode \dot{}\else \.{}\fi{}(t)] and (1/\ensuremath{\surd}2\ensuremath{\omega})[d/dt,\ensuremath{\partial}/\ensuremath{\partial}B${\ifmmode \dot{}\else \.{}\fi{}}^{\mathrm{*}}$(t)], respectively, where B\ifmmode \dot{}\else \.{}\fi{}(t) is a Gaussian white noise and B${\ifmmode \dot{}\else \.{}\fi{}}^{\mathrm{*}}$(t) is a time-reversed Gaussian white noise. The operators were shown to have proper commutation relations and to provide a Hamiltonian which has an eigenvalue equation desired in the {N} representation as should exist in conventional quantum mechanics. Therefore, the present formulation provides a correspondence between stochastic mechanics and conventional quantum mechanics.
IntroductionLet
at and
at∗ be respectively annihilation and creation operators at apoint
t∈R .In the theory of white noise operators [23][24], it is known that
at is acontinuous linear operator from
Γu(Ec) , the (Fock) space of test functions over acomplexified nuclear space
Ec , into itself, and at is acontinuous linear operator from
Γu(EC)∗ , the dual space of
Γu(EC) , into itself.In paticular,
at+at∗ and
at+at∗+at∗at+I are cffied the quantum Gaussian white noise and the quantum Poisson white noise, respectively (cf.[11] [24]).These are known to be elemental generalzed qunatum stochastic processes.The main purpose of this work is to examine relationships between classical and quantum white noises from the point of the classical white noise theory of Gaussian [19][20] and Poisson types [13].Then it will be clarified that quantum Gaussian and Poisson white noises are the Fock space realzations of classical Gaussian and Poisson white noises, respectively.We shall discuss them in Section 4. As amatter of fact, one can see that such facts can be proved not only through
J -transform given by (2.4) and the holomorphy, but also through Sx-transform gllven by (3.3) depending on the exponential function
ϕξX .The function
ϕξX determines aunitary isomorphism between Boson Fock space and
L2(E∗,μx) ,
X=G,
P , where
μG and
μP are a Gaussian measure and aPoisson measure on
E∗ , respectively.As is pointed out in Remark 4.3, the choice of
ϕξX (kernel function of
SX -transform)is the essential part to represent the above different quantum noises acting on the same space
Γu(Ec)∗ .In Appendix, some connections between
[6][7] and [10]
wiU be noted.
We develop a stochastic integration theory with respect to creation, annihilation and gauge operators on the full Fock space. This is done by using a kernel representation for a large class of bounded operators on the full Fock space. It is shown that the kernels form a Banach algebra. Having established the definition of processes and stochastic integrals we go on to prove an Ito formula and use this for examining stochastic evolutions and constructing dilations of special completely positive semigroups. Explicit solutions of the corresponding stochastic differential equations are given.
A review of the second quantization formulation of quantum mechanics is given based on the idea of a graded Hilbert space whose elements are represented by column matrices containing zero, one, two, ···, particle functions. The creation and annihilation operators are defined in terms of intuitive ideas about adding a particle to or removing a particle from the system. The (anti)commutation relations are shown to follow from the (anti)symmetry of the wavefunctions. The field operators which describe the creation or annihilation of a particle at a point are then obtained and their properties discussed. These operators are then used to express general operators in second quantization. Finally, a basis is introduced and the equations of occupation-number space are obtained.
By virtue of stochastic decoupling, the quantum evolution of a harmonic oscillator is converted into two virtual contributions dictated by either a stochastic creation operator or a stochastic annihilation operator. Two approaches are provided to resolve these virtual two evolution operators. One is based on the coherent-state representation and the other deals with the momentum representation after separating out the position and momentum factors. The quantum propagator of the harmonic oscillator is obtained by taking the random average of the combined virtual propagators. The main mathematical exercise of the whole procedure is the calculation of the Gaussian integral.
The properties of a quantum mechanical system composed of many identical particles are most conveniently described in terms of the second-quantized, Heisenberg representation, particle-creation, and annihilation operators. The creation operator, ψ †( r , t), when acting to the right on a state of the system, adds a particle to the state at the space-time point r , t; the annihilation operator ψ ( r , t), the adjoint of the creation operator, acting to the right, removes a particle from the state at the point r , t.