The free wavicle: motivation for the Schrödinger Equation
• Einstein showed that hitherto wavelike phenonomenon had
distinctly particle-like aspects: the photoeffect
• photon energy is E = hf = ħw (h = Planck’s constant; f =
frequency; w = angular frequency = 2pf)
• let f(x,t) be a wave’s amplitude at position x at time t 2
2
 f
2 f
• the Classical Wave Equation reads (v = phase speed) 2  v
2
t
x
• first pass at a solution is any function in the form f(± x – vt); a
pattern that moves at speed v to R (+) or to L (–) with fixed shape
• can build linear combinations that satisfy CWE with different
phase speeds, too, so the pattern may in fact evolve as it moves
• consider the simple harmonic solution [wavelength l; period T
 2p 

2p 
 2p
f ( x, t )  A cos 
x
t   A cos kx  wt 
 x  vt   A cos 
T 
 l
 l 

2p
2p
l w
where wavenumber k 
; angular frequency w 
; phase speed v  
l
T
T k
Going to a complex harmonic wave
• sines are as good as cosines, so if we take a complex
sum as follows, it also works (assume A is real):
i(kx  wt )
f ( x, t )  Acos kx  wt   i sin  kx  wt   Ae
so we see Re  f   A cos kx  wt  and Im  f   A sin  kx  wt 
• the real and imaginary parts are 90° out of phase
• in the complex plane, for some x, f orbits at w CW on a circle
of radius A  much simpler than ‘waving up and down’
• this will be the frequency of the quantum oscillations
Einstein on waves  de Broglie on particles
hw
photon energy E  hf 
: w
2p
relativity
E 2  p 2 c 2  m02 c 4 but mass is zero  E  pc
w
h
h
 p
 k   de Broglie wavelength l 
c
l
p
Non-relativistic wavicle physics
• free Non-Relativistic Massive Particle has
• non-free NRMP has
p 2  2k 2
EK

2m
2m
p2
 2k 2
E  K  V ( x) 
 V ( x) 
 V ( x)
2m
2m
• we connect this energy to the photon energy
 2k 2
 V ( x)  w
2m
• let the wavicle amplitude function be written Y(x,t) and for a
free wavicle we take the earlier complex form Y( x, t )  Aei(kx  wt )
Y
 2Y
2
 note
 iwY (fails for sine/cosin e form) and


k
Y (works)
2
t
x
 2k 2
 the
NRMP energy equation, with Y inserted :
2m
Y ( x, t )  V ( x ) Y ( x , t )  E Y ( x , t )
  2  2 Y ( x, t )
Y ( x, t )
 Time - Dependent Schrodinge r Equation

V
(
x
)
Y
(
x
,
t
)

i

2m
t
x 2
• compare to classical wave equation (order, reality..)
What is this thing, the wavefunction Y(x,t)?
• it contains information about physics: position, momentum,
kinetic energy, total energy, etc. using operators

pˆ  i
x
xˆ  x
2
2
2
ˆ
p



Kˆ 

2m
2m x 2
2
2



Eˆ : Hˆ 
 V ( x)
2
2m x
• we assume that TDSE also works for a non-free particle if
energy is conserved (V = V(x) so its operator is trivial)
• Born (Max) interpretation of complex wavefunctionY(x,t)
-- Y*Y = r(x,t) probability density at time t; Y = probability amplitude
-- Y*Y dx = probability that, at time t, particle is between x and x + dx
b

Pab  Y * Ydx  probabilit y that particle is in interval a  x  b, at time t
a

1
 Y * Ydx  wavefunction is said to be normalized at every time


 Q :

Y * (Qˆ Y )dx one ' sandwiches' the operator and it acts to the right

Some peculiarities of the free wavicle f(x,t)
• it exists finitely everywhere so does not represent a ‘bound state’
2mE
k 2
i ( kxwt )
i ( kxwt )
f ( x, t )  Ae
so f * ( x, t )  Ae
where k 
;w

2m
 f * f  A2 so the probabilit y is constant at any time at any place!!


f*f
dx   so the free wavicle is not normalizab le in this fashion!
-
 vphase : vquantum 
w
k

k 2
2m
k

k
2 E k
whereas vgroup : vclassical 

2m
m
m
• we will revisit these ideas again but we need a lot more insight
into the subtle distinctions between bound states (which are
discrete) and free states (which form a continuum)
• for now, the normalization integral IS infinite but it will turn out
that a free wavicle with any other wavenumber is orthogonal – so
the infinity is really a dirac delta function in ‘k-space’
Mathematical attributes of Y
• it must be ‘square-integrable’ over all space, so it has to die off
sufficiently quickly as x ± ∞, to guarantee normalizability
• the free wavicle fails this test! Normalizing it is tricky!
• no matter how pathological V(x), Y is piecewise continuous in x
• let’s check whether Yx,t ‘stays’ normalized as time goes by..
d
dt




Y * Ydx 





Y * Y dx [since the boundary is not changing with time ]
t
Y 
 Y *
Y

Y
*
 t
 dx

t




this thing should be zero!!
  2  2 Y ( x, t )
Y ( x, t )
TDSE reads

V
(
x
)
Y
(
x
,
t
)

i

2m
t
x 2
Y i  2 Y iV


 Y
2
t 2m x
h
  2  2 Y * ( x, t )
Y * ( x, t )
take c.c :

V
(
x
)
Y
*
(
x
,
t
)


i

2m
t
x 2
Y *  i  2 Y * iV



Y*
2
t
2m x
h
Finishing the normalization check of the solution to the TDSE
postmultip ly first by Y*, premultipl y second by Y , and add...
V terms cancel and one gets
Y
Y * i   2 Y
 2 Y * 
Y * Y

Y*
Y
2
2


t
t
2m  x
x

compare to previous expression and x - integrate
d
dt



i  Y
Y * 
i
Y * Ydx 
Y
*

Y


2m  x
x
   2m


 Y Y * Y * Y 


 dx
x x 
 x x


• First term is zero because Y has to die off at x = ±∞
• Second term is obviously zero
• therefore, probability is conserved
• a subtle point is that when a matter wave encounters a barrier that
it can surmount, one must consider the probability flux rather than
the probability…
Elements of the Heisenberg Uncertainty Principle
• uncertainty in a physical observable Q is standard deviation sQ
• for the familiar example of position x and momentum p: a particle
whose momentum is perfectly specified is an infinitely long wave,
so its position is completely unknown: it is everywhere!
• a particle which is perfectly localized, it turns out, must be made
of a combination of wavicles of every momentum in equal
amounts, so knowledge of its momentum is lost once it is ‘trapped’
• Heisenberg showed that the product of the uncertainties could not
be less than half of Planck’s constant:
s xs p 

2
• it is amusing to confirm this inequality for well-behaved Y
• there is also an energy-time HUP of the same form, and an
angular momentum-angular position one of the same form
• we’ll derive this soon much more rigorously