The total energy of matter related to the frequency ν of the wave is E=hν the momentum of matter related to the wavelength λ of the wave is p=h/λ 3.1 Matter

the total energy of matter related to the frequency ν of the wave is

E=hν

the momentum of matter related to the wavelength λ of the wave is

p=h/λ

3.1 Matter wave: de Broglie

Ex: (a) the de Broglie wavelength of a baseball moving at a speed v=10m/s, and m=1kg. (b) for a electron K=100 eV.

)baseball()electron(

2.1106.1100101.92/106.6

2// (b)

106.6106.6)101/(106.6/ (a)

193134

253534

o

o

A

mKhph

Amph

Chapter 3 de Broglie’s postulate: wavelike properties of particles


The experiment of Davisson and Germer

(1) A strong scattered electron beam is detected at θ=50o for V=54 V.

(2) The result can be explained as a constructive interference of waves scattered by the periodic arrangement of the atoms into planes of the crystal.

(3) The phenomenon is analogous to the Bragg-reflections (Laue pattern).

1927, G. P. Thomson showed the diffraction of electron beams passing through thin films confirmed the de Broglie relation λ=h/p. (Debye-Scherrer method)

Bragg reflection:


constructive interference:

)wavelength (electron

65.1106.154101.92/106.6

2/

eV 54 electronfor

)wavelength ray-(X

65.165sin91.02sin2

1for

652/90 and 50 , 91.0

sin2

193134o

oo

oooo

A

mKh

K

Ad

n

Ad

nd

consistent


Debye-Scherrer diffraction

X-ray diffraction: electron diffraction : zirconium oxide crystal gold crystal

Laue pattern of X-ray (top) and neutron (bottom) diffraction for sodium choride crystal

3.2 The wave-particle duality


Bohr’s principle of complementarity: The wave and particle models are complementary; if a measurement proves the wave character of matter, then it is impossible to prove the particle character in the same measurement, and conversely

Einstein’s interpretation: for radiation (photon) intensity

is a probability measure of photon density

)/1( 220 NNhcI

2

Max Born: wave function of matter is just as satisfies wave equation

is a measure of the probability of finding a particle in unit volume at a

given place and time. Two superposed matter waves obey a principle of

superposition of radiation.

),( tx

2


3.3 The uncertainty principle

Heisenberg uncertainty principle: Experiment cannot simultaneously determine the exact value of momentum and its corresponding coordinate.

2/

2/

tE

xpx

Bohr’s thought experiment2/2

)/(/

2/22/ (2)

2/2sin/sin)/2(

)/ ( sin/

sin)/2(sin2 (1)

2

''

'

''

hxptE

ptxEtxvtvx

pvmppEmpE

hhxp

ax

hpp

x

xxx

xxxxx

x

x

Bohr’s thought experiment:

a diffraction apparatus


3.4 Properties of matter wave

wave propagation velocity:

Why? 2

2/)()(

2

vwv

mv

mv

p

E

h

E

p

hw

a de Broglie wave of a particle

)(2sin),(

1/set )/(2sin),(

txtx

txtx

(1) x fixed, at any time t the amplitude is one, frequency is ν.

(2) t fixed, Ψ(x,t) is a sine function of x.

(3) zeros of the function are at

these nodes move along x axis with a velocity

it is the node propagation velocity (the oscillation velocity)

tnxtnx

nntx

nn

n

)/(2/2/

,.......2,1,0 )(2

// dtdxw n

modulate the amplitude of the waves


2/

2/ is wave the of velocity group the (2)

/ is wave individual the of velocity the (1)

)(2sin]22

cos[2 ),(2 and 2for

]2

)2(

2

)2([2sin]

22cos[2),(

])()[(2sin),( ),(2sin),(

),(),(),(

21

21

d

dν

dκ

dνg

w

txtd

xd

txdd

td

xd

td

xd

tx

tdxdtxtxtx

txtxtx

group velocity of waves equal to moving velocity of particles

vg

vdp

dEmvdvdEmvpmvE

dp

dE

hdp

hdE

d

dg

h

dpd

h

p

h

dEd

h

E

2

1

/

/

1

2


The Fourier integral can prove the following universal properties

of all wave. 4/1 and ,1/for 4/1 tx

:wavematter for hphp //1/

2/

)/1()/(/

2/

4/1)/1()/(

tE

EthhEthEhE

xp

pxhhpxx

uncertainty principle

uncertainty principle

the consequence of duality

Ex: An atom can radiate at any time after it is excited. It is found that in a typical case the average excited atom has a life-time of about 10-8 sec. That is, during this period it emit a photon and is deexcited. (a) What is the minimum uncertainty in the frequency of the photon? (b) Most photons from sodium atoms are in two spectral lines at about . What is the fractional width of either line, (c) Calculate the uncertainty in the energy of the excited state of the atom. (d) From the previous results determine, to within an accuracy , the energy E of the excited state of a sodium atom, relative to its lowest energy state, that emits a photon whose wavelength is centered at


o

A5890?/ E

eV 1.2)//(//hh/ (d)

state the of width theeV 103.3104

1063.6

4

4/ (c)

line spectral the of width natural 106.1101.5/108/

sec 101.5105890/103/ (b)

sec 1084/14/1 (a)

88

34

8146

-114810

-16

EEEEt

h

t

hE

c

tt

E o

A 5890

uncertainty principle in a single-slit diffraction

for a electron beam:


2

sin

sin ,sin

hyy

hyp

y

pppp

p

p

y

y

yy

y


Ex: Consider a microscopic particle moving freely along the x axis. Assume that at the instant t=0 the position of the particle is measured and is uncertain by the amount . Calculate the uncertainty in the measured position of the particle at some later time t.

0x

xtxx

xmtvtx

xmmpv

xp

x

xx

x

or

2/t timeAt

2//

2/0tAt

0

0

0

0

Dirac’s relativistic quantum mechanics of electron: 420

22 cmpcE

Dirac’s assumption: a vacuum consists of a sea of electrons in

negative energy levels which are normally filled at all points in space.

Some consequences of the uncertainty principle:

(1) Wave and particle is made to display either face at will but not both

simultaneously.

(2) We can observe either the wave or the particle behavior of radiation;

but the uncertainty principle prevents us from observing both together.

(3) Uncertainty principle makes predictions only of probable behavior of

the particles.


The philosophy of quantum theory:

(1) Neil Bohr: Copenhagen interpretation of quantum mechanics.

(2) Heisenberg: Principally, we cannot know the present in all details.

(3) Albert Einstein: “God does not play dice with the universe”

The belief in an external world independent of the perceiving subject is

the basis of all natural science.

Documents

The total energy of matter related to the frequency ν of the wave is E=hν the momentum of matter related to the wavelength λ of the wave is p=h/λ 3.1 Matter