Quantum Mechanics
Schrodinger equation
Quantum Mechanics
Propagators : Pg
Quantum Simple Harmonic Oscillator QSHO
Quantum Mechanics
Simulation With GNU Octave
© The scientific sentence. 2010

 Expectation
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 ε_{o} c E^{2},
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 E^{2}. 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

