Interference

Interference is the superposition of two or more waves resulting in a new wave pattern. As most commonly used, the term usually refers to the interference of waves which are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency. Two non-monochromatic waves are only fully coherent with each other if they both have exactly the same range of wavelengths and the same phase differences at each of the constituent wavelengths.

The principle of superposition of waves states that the resultant displacement at a point is equal to the sum of the displacements of different waves at that point. If a crest of a wave meets a crest of another wave at the same point then the crests interfere constructively and the resultant wave amplitude is greater. If a crest of a wave meets a trough then they interfere destructively, and the overall amplitude is decreased.

Interference is involved in Thomas Young's double-slit experiment where two beams of light which are coherent with each other interfere to produce an interference pattern (the beams of light both have the same wavelength range and at the center of the interference pattern they have the same phases at each wavelength, as they both come from the same source). More generally, this form of interference can occur whenever a wave can propagate from a source to a destination by two or more paths of different length. Two or more sources can only be used to produce interference when there is a fixed phase relation between them, but in this case the interference generated is the same as with a single source; see Huygens' principle.

Light from any source can be used to obtain interference patterns, for example, Newton's rings can be produced with sunlight. However, in general white light is less suited for producing clear interference patterns, as it is a mix of a full spectrum of colours, that each have different spacing of the interference fringes. Sodium light is close to monochromatic and is thus more suitable for producing interference patterns. The most suitable is laser light because it is almost perfectly monochromatic.

Constructive and destructive interference

When two waves superimpose, the resulting waveform depends on the frequency (or wavelength) amplitude and relative phase of the two waves. If the two waves have the same amplitude

A

and wavelength the resultant waveform will have amplitude between 0 and

2 A

depending on whether the two waves are in phase or out of phase.

combined waveform
wave 1
wave 2
	Two waves in phase	Two waves 180° out of phase

Consider two waves that are in phase,with amplitudes $A_1$ and $A_2$ . Their troughs and peaks line up and the resultant wave will have amplitude $A = A_1 + A_2$ . This is known as constructive interference.

If the two waves are 180° out of phase, then one wave's crests will coincide with another wave's troughs and so will tend to cancel out. The resultant amplitude is $A = |A_1 - A_2|$ . If $A_1 = A_2$ , the resultant amplitude will be zero. This is known as destructive interference.

General Quantum Interference

Constructive and destructive interference are examples of a more general form of quantum interference.

If a system is in state $\psi$ its wavefunction is described in Dirac or bra-ket notation as:

$|\psi \rang = \sum_i |i\rang \psi_i$

where the $|i\rang$ s specify the different quantum "alternatives" available (technically, they form an eigenvector basis) and the $\psi_i$ are the probability amplitude coefficients, which are complex numbers.

The probability of observing the system making a transition or quantum leap from state $\psi$ to a new state $\phi$ is the square of the modulus of the scalar or inner product of the two states:

$prob(\psi => \phi) = |\lang \psi |\phi \rang|^2 = |\sum_i\psi^*_i \phi_i |^2 = \sum_{ij} \psi^*_i \psi_j \phi^*_j\phi_i= \sum_{i} |\psi_i|^2|\phi_i|^2 + \sum_{ij;i \ne j} \psi^*_i \psi_j \phi^*_j\phi_i$

where $\psi_i = \lang i|\psi \rang$ (as defined above) and similarly $\phi_i = \lang i|\phi \rang$ are the coefficients of the final state of the system. * is the complex conjugate so that $\psi_i^* = \lang \psi|i \rang$ etc.

Now let's consider the situation classically and imagine that the system transited from $|\psi \rang$ to $|\phi \rang$ via an intermediate state $|i\rang$ . Then we would classically expect the probability of the two-step transition to be the sum of all the possible intermediate steps. So we would have

The classical and quantum derivations for the transition probability differ by the presence, in the quantum case, of the extra terms $\sum_{ij;i \ne j} \psi^*_i \psi_j \phi^*_j\phi_i$ ; these extra quantum terms represent interference between the different $i \ne j$ intermediate "alternatives". These are consequently known as the quantum interference terms, or cross terms. This is a purely quantum effect and is a consequence of the non-additivity of the probabilities of quantum alternatives.

The interference terms vanish, via the mechanism of quantum decoherence, if the intermediate state $|i\rang$ is measured or coupled with the environmentWojciech H. Zurek, Decoherence and the transition from quantum to classical, Physics Today, 44, pp 36-44 (1991)Wojciech H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Reviews of Modern Physics 2003, 75, 715 or ^*.

References

External links

For expressions of position and fringe spacing click the link below: http://www.citycollegiate.com/interference1.htm

Java demonstration of interference

Interference | Physical optics | Wave mechanics

Gourt Home::Gourt :: Interference

Constructive and destructive interference

General Quantum Interference

See also

References

External links