Coherent state
Coherent States ( see also coherent radiation ) are quantum mechanical states indefinite number of particles, mostly in bosons.
As R. J. Glauber showed in 1963, can the electromagnetic wave of a laser mode is best described by coherent states.
Coherent states are classical electromagnetic waves very close because of the expectation value of the electric field has the form of a classical electromagnetic wave, regardless of the expected value of the number of particles.
If you measure in a coherent state, the number of particles in each case in a fixed time interval, we obtain values that satisfy a Poisson distribution.
Description in Fock space
An ideal coherent state in quantum field theoretical treatment of photons, electrons, etc. is always a superposition of states of different particle number, it contains even ( vanishingly small ) shares of arbitrarily large number of particles. In Fock space notation (after Vladimir Aleksandrovich Fock ), the coherent state results as an infinite linear combination of states of fixed particle ( Fock states ) to:
This is an arbitrary non-vanishing complex number which completely defines the coherent state. The probability to measure a cast of exactly n particles
The distribution therefore corresponds to the Poisson distribution. It follows that the expectation value of the occupation number of the coherent state. The coherent state can be created easily by use of a unitary " shift operator " of the unoccupied state of the system (see section derivation )
In this case, and the up and lowering operators of the Fock state.
Properties
Important properties of a coherent state are:
- Normalization: The pre-factor of the coherent state is thus used the normalization.
- Orthogonality: Coherent states are not orthogonal.
- The coherent state is a " right-sided " eigenstate of the annihilation operator. The rule is:.
- Coherent states have minimal blur:
- In a non-interacting theory ( the harmonic oscillator ) coherent states remain coherent. However, they are not eigenstates of the free Hamiltonian. Rather, the phase rotates at the oscillator frequency, that is, A coherent state is transferred to another consistent state.
In quantum electrodynamics, the coherent state is an eigenstate of the operator of the vector field (or, equivalently, the electric field ). In a coherent state, the quantum fluctuations of the electric field is identical to that of the vacuum state.
History
The above-mentioned consistent state has been identified by Erwin Schrödinger, when this searching for a state of the quantum harmonic oscillator, which corresponds to that of the conventional harmonic oscillator, and transferred by Roy J. Glauber the Fock space. The coherent state thus corresponds to a Gaussian wave packet which reciprocates in the harmonic potential, without changing location or momentum uncertainty.
Derivation
In the following, we show that the coherent states are eigenstates of the annihilation operator:
The Fockzustände form a complete ortho- normal system, so you can develop each state according to them:
Now if one considers the left side of the eigenvalue equation, where and:
Mixing up and the infinite sum (and thus a limit education) is by no means trivial, because even in the case of the harmonic oscillator, a discontinuous operator. In the case of the harmonic oscillator can justify this step, in general, caution is necessary here, however!
The right-hand side of the eigenvalue equation:
From the equality of both sides to win a recurrence relation
Now one uses the normalization condition of the coherent states in order to determine:
Square root yields, with a complex phase to zero and is thus chosen real:
This is substituted into the above development, the representation of coherent states:
One uses more of that Fockzustände be found by applying the generating operator of the vacuum state and then that application of the annihilation operator produces the vacuum state or a zero, then we obtain:
Using the Baker -Campbell - Hausdorff formula can be summed the product of two exponential functions, where:
Thus
Coherent states in quantum mechanics
In the one -particle quantum mechanics is meant by a coherent state is a Gaussian wave packet with real variance. As additional parameters, it has the expected value q of the place and the expectation value p of the pulse. The normalized wave function in position space is in one space dimension
The corresponding ket is defined by
Quasi- Classical Properties
The blurring of position and momentum are given by the Gaussian wave packet
Thus, the uncertainty product assumes the minimum value:
The reverse is also follows from a minimum uncertainty product, that the wave function is a Gaussian wave packet.
In the limit, the wave packet to an eigenstate of the place, in the limit to an eigenstate of the pulse. By " classical" conditions when both can be considered as small, the Gaussian wave packet is approximately a common eigenstate of the position operator and momentum operator:
The errors are of the order of magnitude of the blur, because as a measure of the deviation of the eigenvalue equation can just apply the expressions that define the uncertainties (see above).
Completeness relation
Each wave packet can be represented as a superposition of Gaussian wave packets. As an operator equation formulated ( partition of unity ):
This can show, by forming on either side of the array element in the local base and on the right side uses the wave function and the Fourier representation of the delta function.
Planck's constant is in this way for a reference for classical phase volumes.
Application: Classical partition function
Using the completeness relation, the classical one -particle partition function for the canonical ensemble can easily be obtained from the quantum mechanical partition function
Are derived. Namely, if the uncertainties of position and momentum is negligible and thus the coherent states are common eigenstates of position, momentum and Hamiltonian applies
Was being used. A more detailed argument with upper and lower bounds can be found in.