The 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 of finding a particle in the region [x, x+ dx].

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

<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 of locating a particle in region [x; x+dx] is |y(x)|2dx. Since the coordinate changes continuously, the average of distribution sum  goes into the 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 average momentum of distribution is calculated in principally the same way that we considered before but now we do not need to know the statistical distribution |y(p)|2

<p>  = -∞∫+∞(py(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 of the mean momentum value of a 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 seems amazing: how can the mean impulse of particle located at Δx be zero if the Heisenberg uncertainty principle states that each particle should have non-zero momentum? The answer: <p> is the mean impulse rather than instantaneous one. The particle moves forward 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-squared deviation <Dp2> of the mean impulse in order to obtain the range of impulse variation. It is advised for everyone to do this.

Auf diesem Webangebot gilt die Datenschutzerklärung der TU Braunschweig mit Ausnahme der Abschnitte VI, VII und VIII.