So, let's say it again that it's principally impossible to write down fundamental principles in form of basic equations. One can (and surely must) try to draw conclusions using heuristic way of thinking and then try to compare a number of solutions of "invented" equation with experimental measurements.
We have seen earlier it's possible to describe observations using sum of probability amplitude. One can write down the following expression for one-dimensional wave:
φ = A e−iwt+ikx where k = 2π/λ
according to de Broglie we can write down the impulse p: λ
= h/p and energy E = hν = hw
of a particle. Substituting expressions for impulse and energy in the first
equation and replacing φ
by the probability amplitude
y one can obtain the following expression:
y = Ae−iE/ht+ip/hx
¶y/∂t
= -iE/ |
® | (i |
¶y/∂x
= ip/ |
® | ( |
One can see that there are E and p on the right sides of either equation and
they certainly can be measured in experiments. On the left side of each
expression there are kind of instruction how one can obtain energy or impulse:
differentiating y on x (and multiplying it by
ih) one can obtain Ey;
differentiating y on x (and multiplying it by
h/i) one can
obtain py. This mathematical instructions ate also
referred to as operators.
We have introduced them at the end of last chapter but now we can draw a
parallel between physical quantities and operators:
Energy | E | → | i |
Energy operator ≡ H |
Impulse | p | → | Impulse operator ≡ p | |
Coordinate | x | → | x | Characterizing operators we use "bold" symbols |
Classical | Operator |
p = (px, py, pz) | p = |
E = p²/2m + V | H = − |
Lx = ypz − zpy | Lx = |
Ly = zpx − xpz | Ly = |
Lz = xpy − ypx | Lz = |
Δ is the Laplace operator: Δ
= ∂²/∂x²
+ ∂²/∂y²
+ ∂²/∂z².
p²/2my + V(x)y = Ey
and then expand expression for impulse using operator conception p²y
= p(py) = p·(h/i·∂/∂xy)
= − h² (∂/∂x)(∂/∂x)y
= - h² (∂²/∂x²)y,
and also use energy operator ih ∂/∂t,
one can obtain the following
time dependent Schroedinger
equation:
[− |
Then let's try to spread the Schroedinger equation on three dimensions x, y, z:
p = (px,
py, pz) →h/i(∂/∂x,
∂/∂y,
∂/∂z)
p. p
= p2 = (px2
+ py2 + pz2)→
− h2(∂²/∂x²
+ ∂²/∂y²
+ ∂²/∂z²)
= − h2Δ
[− |
y = i |
½¾¾¾¯¾¾½ | |
H : | Hamilton operator |
time dependent Schroedinger
equation:
H y=
i |
If potential (or generally H)
doesn't depend on time and energy E is almost independ of time we can get rid of
time here in the following way:
y(x,y,z,t) =
yu(x,y,z)
e−iE/ |
time independent Schroedinger
equation:
Hyu = Eyu |
This ODE gives us values of stationary energy states.
If we know the time independent solution yu
we can readily write down the solution of time dependent equation unless
H
is not a function of t and yu
describe stationary time independent energy states E:
y =yue−iE/ |
The normalization condition:
-¥ò+¥|y(x,y,z)|2dx dy dz = 1 |
The expression |y(x,y,z)|2dxdydz
gives us the possibility P(x,y,z) to detect particle in the location (x,y,z)
with the range [x,x+dx], [y,y+dy] and [z,z+dz]. P(x,y,z) has unlike
y(x,y,z) clear physical meaning of probability. The
probability amplitude describe considered system completely and one can obtain
all information about the system according to probability amplitude.