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1. Probability in Quantum Mechanics

Max Born (1926) had clarified how a probability 
of finding  a quantum system (particle) in unit volume 
at a certain time is related to its wave function. If ψ(x,t) is the 
wave function of the particle, then the probability density to 
find this particle within a unit volume dV at an instant of time 
"t" is given by |ψ|2.
We can inderstand this result from the fact that for an 
electromagnetic radiation (light) the intensity 
(energy flux) is εo c E2, 
where E is the electric field of the radiation. As the 
amplitude of the wave function ψ is the electric field 
E, |&psi|2 is proportional to E2. The 
more the intensity is large, the more the particle is likely 
to be seen.

Let a particle constrained to move in one dimention 
space (x). Its wave function is ψ(x,t) at a time "t". The 
probability to find this particle between x and x + dx is: 

P(x) dx = ψ(x,t)ψ(x,t)* dx =  |ψ(x,t)|2 dx 

The probability to find this particle between x1 and x2 is: 
P = ∫ ψ(x,t)ψ(x,t)* dx  [from x1 to x2]

The probability to find this particle anywhere is: 
P = ∫ ψ(x,t)ψ(x,t)* dx = 1  [from - ∞ to ∞]

2. Expectation value for any observable

First, let's measure a quantity x as a position of 
a particle. If we get n1 times the value x1 , n2 times the value x2, 
..., and ni times the value xi, we can evaluate the 
average value of x by writing in descrete notation: 
x = Σ ni xi / Σ ni     i = 1, 2, 3, ...  n 
(xi are the results of the measurement of "x", ni is the number 
of the results "xi", "n" is the numer of the results, nad Σni = N 
is the total number of the measurements.)

ni/N = ni/Σ ni is the probability to have the value xi. If we 
denote this probability by Pi, we can write:

x = Σ ni xi / Σni =  Σ (ni/N) xi / Σ (ni/N) = Σ Pi xi / Σ Pi
Normalized to 1, we have Σ Pi = 1. Therefore:
x  =  Σ Pi xi     i = 1, 2, 3, ...  N 

In continuous notation, we have: 
x = ∫ P(x)dx x /∫ P(x)dx [-∞ + ∞]. 

In Quantum Mechanics, the wave function is used to 
determine the expectation value of any physical observable:

For the Observable position x , the expectation value is:
<x> = ∫ P(x) x dx / ∫ P(x)dx [-∞ + ∞] = 
∫ ψ(x,t) x ψ(x,t)* dx /∫ P(x)dx [-∞ + ∞]

Normalized to 1, that is: ∫ P(x)dx [-∞ + ∞] = 1, we get:
<x> = ∫ P(x) x dx 

In the general case:

For the physical observable A, of the normalized wave 
function ψ(r,t), the expectation value of A within 
the volume element dV is given by:
<A> = ∫P(r)Adr=∫ψ(r,t)Aψ(r,t)*dV


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Combinatorics - Probability

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