Averages of Distribution are Mean Values

Averages of Distribution:

Using y one can calculate the mean values (the mean value is called the average of distribution in quantum mechanics) of a measurement; for instance, the probability to detect particle in the region [x, x+ dx].

The average of distribution in statistics is the sum of single measured value multiplied by probability to which corresponds the measured value.

<x> | = | Σi | xi | Pi
|  |  |  | Pi: probability that the measured value is xi
|  |  | i: index of xi
<x>: mean value of measured values xi

Coordinate mean value (one-dimensional):

The probability to detect particle in region [x; x+dx] is |y(x)|2dx. Since the the coordinate changes continuously then the average of distribution sum goes into integral:

<x> = -∞∫+∞ x |y(x)|2dx = -∞∫+∞y*(x) x y(x) dx

y(x) is normalized in the following way -∞∫+¥ |y|2dx = 1.

Generally it will be the following:

<xn> = -∞∫+¥ y*(x) xn y(x)dx

Example of particle in potential well: y1(x) = (2/a)½sin(πx/a)

<x> = o∫a(2/a)½sin(πx/a)· x · (2/a)½sin(πx/a)dx = 2/a o∫ax ·sin2(πx/a)dx

<x> = a/2

Impulse mean value:

The impulse average of distribution is calculated principally in the same way as we've considered before but we must have known also statistical distribution |y(p)|2 beforehand. Now we know the operator sign for impulse, i.e. we can write down:

<p> = -∞∫+∞(p y(x))y*(x)dx = -∞∫+¥ h/i(∂/∂xy)·y*dx

<p> = -∞∫+¥ y* ·(h/i.∂/∂x)y dx

<pn> = -∞∫+¥ y* ·(h/i.∂/∂x)ny dx

The kinetic energy average of distribution (Ekin = p²/2m):

<Ekin> = -∞∫+¥ y* (−h²/2mΔ)y dx

Δ = ∂²/∂x² + ∂²/∂y² + ∂²/∂z²

Example for mean impulse value of particle in a well: y1(x) = (2/a)½sin(πx/a) ; ¶y1/∂x = π/a(2/a)½cos(πx/a)

<p> = o∫a(2/a)½ sin(πx/a).h/i.π/a·(2/a)½.cos(πx/a) = 0

At first this result amazest: how can the mean impulse of particle located at place Δx be zero if there is the Heisenberg uncertainty principle that states each particle should have non-zero impulse? The answer: <p> is the mean impulse rather than momentary one. The particle moves there and back with the same speed and therefore the mean speed (together with mean impulse) is equal to zero. We can also calculate the mean-square deviation <Dp2> of the mean impulse in order to obtain the range of impulse variation. It is advised to be done by everyone.