Coherent and Incoherent Superposition) 769

Superposition Principle (Coherent and Incoherent Superposition) 769

4. R.J. Glauber: Ampliﬁers, attenuators, and Schr¨odinger’s Cat,’ in New Techniques and Ideas in Quantum Measurement Theory, ed. by D. M. Greenberg, Ann. N.Y. Acad. Sci. 480, 336 (1986) 5. N. Herbert: FLASH – A superluminal communicator based upon a new type of quantum measurement. Found. Phys. 12, 1171 (1982) 6. M. D. Stenner, D. J. Gauthier, M. A. Neifeld: The speed of information in a ‘fast light’ optical medium. Nature 425, 695 (2003) 7. W. K. Wootters, W. H. Zurek: A single quantum cannot be cloned. Nature 299, 802 (1982)

Secondary Literature

8. M. Fayngold: Special Relativity and Motions Faster Than Light (Wiley-VCH Verlag GmbH, Weinheim 2002, Sect. 6.15) 9. N. Herbert: Faster Than Light (New American Library Books, Penguin Inc., New York, 1998) 10. P. W. Milonni: Fast Light, Slow Light, and Left-Handed Light (Institute of Physics Publishing, Bristol 2005, Chap. 3) 11. H. Paul: Introduction to Quantum Optics, From Light Quanta to Quantum Teleportation (Cambridge University Press, Cambridge, 2004, Sect. 11.5) 12. J. A. Zensus, T. J. Pearson (eds.): Superluminal Radio Sources (Cambridge University Press, Cambridge, 1987)

Superposition Principle (Coherent and Incoherent Superposition)

Marianne Breinig

In non-relativistic quantum mechanics, the state of a physical system at a fixed time t is defined by specifying a ket |ψ(t) belonging to the space ε. ε is a complex, S separable Hilbert space, a complex linear vector space in which an inner product is defined and which possesses a countable orthonormal basis. Every measurable physical quantity is called an observable and is described by a Hermitian operator acting in ε. The only possible results of a measurement are the eigenvalues of the Hermitian operator associated with the measurement, and immediately after the measurement the state ket is a corresponding eigenstate. Every Hermitian operator has at least one basis of orthonormal eigenvectors. Every state vector |ψ(t) can therefore be written as a linear superposition of eigenvectors of any observable. If two Hermitian operators commute a common eigenbasis can be found. If they do not commute, then no common eigenbasis exists. Let {|an} be an orthonormal basis of eigenvectors of the operator A,

A |an=an| an. (1) 770 Superposition Principle (Coherent and Incoherent Superposition)

For simplicity assume that the eigenvalues are not degenerate. Let |ψ1 and |ψ2 be two normalized eigenvectors of the operator B with eigenvalues b1 and b2, respectively. B|ψ1=b1|ψ1, B|ψ2=b2|ψ2. (2)

If B is the Hamiltonian H, then b1 = E1 and b2 = E2. If A and B do not commute, i.e. [A, B] = 0, then |ψ1 and |ψ2 are linear superpositions of eigenvectors of A. Assume that [A, B] = 0 and that a measurement at t = 0 determines |ψ(0)= |ψ1. If B is the Hamiltonian, then the measurement determines that the system is in a stationary state. The probability that a subsequent measurement of A will 2 yield the eigenvalue an is P1(an) =|an|ψ1| . Similarly, if |ψ(0)=|ψ2 then 2 P2(an) =|an|ψ2| . Now consider a system in a normalized pure state ( states, pure and mixed)

2 2 |ψ=λ1|ψ1+λ1|ψ2, ψ|ψ=1, |λ1| + |λ2| = 1. (3)

If B is the Hamiltonian, then the system is not in a stationary state, it is in a coherent superposition of stationary states. 2 2 The probability that a measurement of B will yield b1 is |ψ1|ψ| =|λ1| .The 2 probability that a measurement of B will yield b2 is |λ2| . The probability that a measurement of A will yield an is

2 P (an) = |an| ψ|

2 2 =an|ψψ |an=| λ1| P1 (an) +|λ2| P2 (an) + ∗ × | | 2Re(λ1λ2 an ψ1 ψ2 an ) 2 2 =|λ1| P1 (an) +|λ2| P2 (an) . (4)

The last term in the expression for P(an) describes interference effects. If a system is in a pure state which is a coherent superposition of eigenstates of an observable B and we measure an observable A which does not commute with B, then we must take interference effects into account when predicting the result of a measurement. 2 We may consider P(an) =|an|ψ| as the square of the probability amplitude an|ψ=an|λ1ψ1+an|λ2ψ2. The probability amplitude is the weighted sum of the probability amplitudes an|ψ1 and an|ψ2. To obtain the probability P(an) for a linear superposition of states, we take the square of the weighted sum of the probability amplitudes, not the sum of the squares. A pure state is not a statistical mixture of states. The concept of a statistical mixture of states ( mixed state) is used when dealing with incomplete information about the initial state of a system. Assume it is only known that the system is in one of the eigenstates {|ψk} of the operator B and that it has the probability = | pk ( k pk 1) of being in the pure state ψk . If B is the Hamiltonian, the system then is in an incoherent superposition of stationary states. If the system is in a sta- 2 2 tistical mixture of the states |ψ1 and |ψ2 with weights p1 =|λ1| and p2 =|λ2| 2 2 respectively, then the probability of measuring an is P(an) =|λ1| P1(an) +|λ2| Superselection Rules 771

P2(an). Interference effects are absent for an incoherent superposition or a statistical mixture of states. We cannot describe a statistical mixture using an “average state vector”. In general, when dealing with a statistical mixture, probabilities enter at two levels. The initial information about the system is given in terms of probabilities, and the predictions of Quantum Mechanics are probabilistic. A simple example: Let the operator B be the Hamiltonian of the system, B = H, b1 = E1, b2 = E2, and let |ψ(0)=λ1|ψ1+λ1|ψ2.Then|ψ(t)=λ1 exp(−iE1t/)|ψ1+λ2 exp(−iE2t/)|ψ2,and

2 2 P (an) = |λ1| P1 (an) + |λ2| P2 (an) + ∗ × − − | | 2Re(λ1λ2 exp( i (E1 E2) t/ ) an ψ1 ψ2 an )

P(an) now is time dependent and oscillates with a frequency ν12 = (E1 − E2)/h. We observe quantum beats.

Primary Literature

1. C. Cohen-Tannoudji, B. Diu, F. Lalo¨e: Quantum Mechanics, Volume One (Wiley, New York 1977)

Secondary Literature

2. P.A.M. Dirac: The Principles of Quantum Mechanics, 4th edn. (Oxford University Press, Oxford 1958)

Superselection Rules S

Domenico Giulini

General Notion

The notion of superselection rule (henceforth abbreviated SSR) was introduced in 1952 by Wick (1909–1992), Wightman, and Wigner (1902–1995) [9] in connection with the problem of consistently assigning intrinsic parity to elementary particles. They understood an SSR as generally expressing “restrictions on the nature and scope of possible measurements”. The concept of SSR should be contrasted with that of an ordinary selec- tion rule (SR). The latter refers to a dynamical inhibition of a certain transition,