In the context of the double-slit experiment Probability amplitude

probability amplitudes have special significance because act in quantum mechanics equivalent of conventional probabilities, many analogous laws, described above. example, in classic double-slit experiment, electrons fired randomly @ 2 slits, , probability distribution of detecting electrons @ parts on large screen placed behind slits, questioned. intuitive answer p(through either slit) = p(through first slit) + p(through second slit), p(event) probability of event. obvious if 1 assumes electron passes through either slit. when nature not have way distinguish slit electron has gone though (a more stringent condition not observed ), observed probability distribution on screen reflects interference pattern common light waves. if 1 assumes above law true, pattern cannot explained. particles cannot said go through either slit , simple explanation not work. correct explanation is, however, association of probability amplitudes each event. example of case described in previous article. complex amplitudes represent electron passing each slit (ψfirst , ψsecond) follow law of precisely form expected: ψtotal = ψfirst + ψsecond. principle of quantum superposition. probability, modulus squared of probability amplitude, then, follows interference pattern under requirement amplitudes complex:

p
=

|


ψ


f
i
r
s
t



+

ψ


s
e
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d





|


2


=

|


ψ


f
i
r
s
t





|


2


+

|


ψ


s
e
c
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d





|


2


+
2

|


ψ


f
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r
s
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|


|


ψ


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|

cos
⁡
(

φ

1


−

φ

2


)
.


{\displaystyle p=|\psi _{\rm {first}}+\psi _{\rm {second}}|^{2}=|\psi _{\rm {first}}|^{2}+|\psi _{\rm {second}}|^{2}+2|\psi _{\rm {first}}||\psi _{\rm {second}}|\cos(\varphi _{1}-\varphi _{2}).}

here,




φ

1




{\displaystyle \varphi _{1}}

and




φ

2




{\displaystyle \varphi _{2}}

arguments of ψfirst , ψsecond respectively. purely real formulation has few dimensions describe system s state when superposition taken account. is, without arguments of amplitudes, cannot describe phase-dependent interference. crucial term



2

|


ψ


f
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r
s
t




|


|


ψ


s
e
c
o
n
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|

cos
⁡
(

φ

1


−

φ

2


)


{\displaystyle 2|\psi _{\rm {first}}||\psi _{\rm {second}}|\cos(\varphi _{1}-\varphi _{2})}

called interference term , , missing if had added probabilities.

however, 1 may choose devise experiment in observes slit each electron goes through. case b of above article applies, , interference pattern not observed on screen.

one may go further in devising experiment in gets rid of which-path information quantum eraser . then, according copenhagen interpretation, case applies again , interference pattern restored.