courses:lecture:wvlec:probprobdens_wiki.ppt
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Transcript courses:lecture:wvlec:probprobdens_wiki.ppt
1
BASICS OF QUANTUM MECHANICS
Reading:
QM Course packet – Ch 5
x
x1 y
is a ket that is the eigenstate of position
is a number that represents the projection of the state
vector onto the ket x1
x2 y
is a number that represents the projection of the state
vector onto the ket x2
xy
y ( x)
x2
x
We’ve represented the general state vector in a
y ( x ) : graphical form by projecting onto position eigenstates.
This the “position representation”. Careful, though …
(x) can be complex, so then we’d have to plot both
the real and imaginary parts for a full representation.
2
y ( x) = x y
y x
Then what is
?
y x = xy
*
= y * ( x)
Then we have the following identifications (not equalities)
y
y ( x)
y
y ( x)
*
3
4
(x) is NOT a physically accessible quantity; we cannot measure
it in the laboratory. The physically meaningful quantity is |(x)|2.
This is the probability density - the probability per unit volume
in 3D, or probability per unit length in 1D) of finding the particle
in an infinitesimally small region located at x.
Ã( x) º y * ( x )y( x) = y ( x )
2
The probability of finding this particle somewhere in the universe
must be 1. This statement is represented by:
¥
¥
-¥
-¥
ò Ã( x) dx = ò y * ( x )y ( x )dx = 1
y y =1
In bra-ket notation:
¥
This suggests that
®
ò
dx
5
Examples ………….
These 1-D wave functions are NOT properly normalized.
Normalize them!
ì
0
x <0
ï æ 3px ö
y ( x ) = ísinç
÷ 0< x<L
ï è L ø
0
x>L
î
ì 0
x < -L /2
ï æ px ö
y ( x ) = ícosç ÷ -L /2 < x < L /2
ï èLø
x > L /2
î 0
We don’t know (yet) how to find wave functions for any
systems – we’ll get there!
6
This the probability density also tells us about the
probability of finding a particle in a certain region of
space, say between x = a and x = b.
b
Ãa< x <b = ò y * ( x )y ( x ) dx
a
Think about the dimensions of the two quantities
Ã( x ) with an argument, x:
dimensions of 1/[L] in 1-D
and of
Ã
with no argument: dimensionless