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The Observer Effect
-------------------
Consider the double slit experiment. Classically we would expect
to see Gaussian distributions produced by each slit that overlap and
sum to produce a single Gaussian of greater amplitude as shown
below.
Instead, though, what we see is an interference pattern as follows.
Mathematically, we can describe this as follows:
| |
| |
slit 1 | ^
| cylinderical | |
ψ(x) -->| waves | | y
| with x and y |
slit 2 components |
| |
|<-------- L -------->|
Let r be the radial distance from a slit.
Slit 1 closed: |ψ_{2}> => exp(ip_{2}r/h)
Slit 2 closed: |ψ_{1}> => exp(ip_{1}r/h)
r = √(L^{2} + y^{2})
Consider the y direction only.
Slit 1 closed: |ψ_{2}> => exp(ip_{2}y/h)
Slit 2 closed: |ψ_{1}> => exp(ip_{1}y/h)
Slits 1 and 2 open: |ψ(y_{12})> = |ψ_{1}> + |ψ_{2}>
_{ } = (exp(ip_{1}y/h) + exp(i_{2}p_{2}y/h))
Probability distribution = ψ^{*}_{12}ψy_{12}
= 2(1 + cos((p_{1}-p_{2})y/h))
This is a periodic function with points of zero value representing
an interference pattern.
If, however, we now introduce a particle counter at each slit that
counts the particles as they pass through, then the pattern disappears
and we again see the classical distribution above. We must conclude
that the act of observation is sufficient to change the state of the
particle and destroy the interference pattern.
The observer effect is not limited to quantum mechanics. The idea
that you cannot measure a system without disturbing it is central to
physics. For example, using a simple thermometer to measure the
temperature of something will slightly alter the system unless the
thermometer is initially at precisely the same temperature as the
system being measured.
The observer effect is subtly different than the Heisenberg Uncertainty
Principle which states that the more precisely the position of some
particle is determined, the less precisely its momentum can be known,
and vice versa. Consider the measurement of a electron's position
using a beam of photons. Measuring the position of the electron with
precision, Δx, requires the wavelength of the photons, λ be less than
Δx. The shorter the wavelength the more precise the measurement will
be. Now, the momentum of a photon is given by:
p = hk = h2π/λ = h/λ
Therefore, shorter wavelength photons have higher momentum. Bombarding
the electron with a high momentum photon causes the electron to move
off in a random direction with an uncertain momentum of order of
magnitude of the incident photon. The result is the momentum of the
electron is much more uncertain than it was before the collision.