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 = ħ (h = Planck’s constant; f = frequency; = angular frequency = 2f)• let f(x,t) be a wave’s amplitude at position x at time t

• 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

• the Classical Wave Equation reads (v = phase speed) 2

22

2

2

x

fv

t

f

• consider the simple harmonic solution [wavelength ; period T

kTv

Tk

tkxAtT

xAvtxAtxf

speed phase;2

frequency angular ;2

r wavenumbewhere

cos 22

cos2

cos),(

Going to a complex harmonic wave

• the real and imaginary parts are 90° out of phase• in the complex plane, for some x, f orbits at CW on a circle of radius A much simpler than ‘waving up and down’• this will be the frequency of the quantum oscillations

p

hhk

cp

pcEcmcpE

hhfE

h wavelengtBroglie de

zero is massbut relativity

:2

energy photon

420

222

Einstein on waves de Broglie on particles

• sines are as good as cosines, so if we take a complex sum as follows, it also works (assume A is real):

tkxAftkxAf

tkxiAetkxitkxAtxf

sin and cos see weso

)(sincos),(

ImRe

Non-relativistic wavicle physics• free Non-Relativistic Massive Particle has m

k

m

pKE

22

222

• non-free NRMP has )(2

)(2

)(222

xVm

kxV

m

pxVKE

• we connect this energy to the photon energy )(

2

22

xVm

k

• compare to classical wave equation (order, reality..)

• let the wavicle amplitude function be written (x,t) and for a free wavicle we take the earlier complex form )(),( tkxiAetx

(works) and form) esine/cosinfor (fails note 22

2

kx

it

t

txitxxV

x

tx

m

txEtxxVtxm

k

),(),()(

),(

2 Equation r SchrodingeDependent -Time

),(),()(),(2

:inserted with equation,energy NRMP the

2

22

22

What is this thing, the wavefunction (x,t)?

• Born (Max) interpretation of complex wavefunction(x,t) -- * = (x,t) probability density at time t; = probability amplitude-- * dx = probability that, at time t, particle is between x and x + dx

• we assume that TDSE also works for a non-free particle if energy is conserved (V = V(x) so its operator is trivial)

right the toactsit andoperator the'sandwiches' one )ˆ(*:

every timeat be tosaid ision wavefunct*1

tat time , intervalin is particley that probabilit *

dxQQ

normalizeddx

bxadxPb

a

ab

• it contains information about physics: position, momentum, kinetic energy, total energy, etc. using operators

)(2

ˆ:ˆ22

ˆˆˆˆ2

22

2

222

xVxm

HExmm

pK

xipxx

• 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’

• it exists finitely everywhere so does not represent a ‘bound state’Some peculiarities of the free wavicle f(x,t)

m

k

m

Evv

m

kkk

vv

dxff

Aff

m

kmEktkxiAetxftkxiAetxf

m

k

2: whereas

2:

fashion! in this lenormalizabnot is waviclefree theso *

!place!any at any timeat constant isy probabilit theso *

2 ;

2 where)(),(* so )(),(

classicalgroup2

2

quantumphase

-

2

2

Mathematical attributes of • 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), is piecewise continuous in x• let’s check whether x,t ‘stays’ normalized as time goes by..

!zero! be should thingis th**

] with timechangingnot isboundary the[since **

dxtt

dxt

dxdt

d

**

2

*

),(*),(*)(

),(*

2 :c.c take

2

),(),()(

),(

2 reads TDSE

2

2

2

22

2

2

2

22

h

iV

xm

i

t

t

txitxxV

x

tx

m

h

iV

xm

i

t

t

txitxxV

x

tx

m

Finishing the normalization check of the solution to the TDSE

• First term is zero because 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…

integrate- and expression previous tocompare

*

*2

**

gets one and cancel terms

add... and ,by secondy premultipl *,by first ly postmultip

2

2

2

2

x

xxm

i

tt

V

dxxxxxm

i

xxm

idx

dt

d

**

2

**

2*

Elements of the Heisenberg Uncertainty Principle• uncertainty in a physical observable Q is standard deviation Q

• 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:

2

px

• it is amusing to confirm this inequality for well-behaved • 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