The Emergence of Quantum Physics

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FI 3103 Quantum Physics

Alexander A. Iskandar

Physics of Magnetism and Photonics Research Group Institut Teknologi Bandung

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Wave Properties of Particle

 Louis-Victor-Pierre-Raymond, 7^th duc de Broglie (15 August 1892 – 19 March 1987) was a French physicist who made ground breaking contributions to quantum theory.

 In his 1924 PhD thesis (thesis advisor : Paul Langevin), he postulated the wave nature of electrons and suggested that all matter has wave properties. This concept is known as wave-particle duality or the de Broglie hypothesis.

Alexander A. Iskandar Emergence of Quantum Physics 3

de Broglie



 h c

ph  de Broglie

wavelength mv

h p h 



 photon momentum

Proof of Wave Properties of Particle

Electron Diffraction

 J. J. Thomson was awarded the Physics Prize in 1906 for showing that electrons are particles. His son, George Paget Thomson, received the same prize in 1937 (together with Davisson) for showing that they also have the properties of waves.

J.J. Thomson

G.P. Thomson C.J. Davisson

C.J. Davisson – L. Germer

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Bragg’s X-ray Diffraction

 Nobel Prize in Physics 1915.

 William Lawrence Bragg, was to date the youngest Nobel Laureate (he was 25 years old when he received the Nobel Prize).

W.H. Bragg W.L. Bragg

Davisson-Germer Experiment

de Broglie relationship

meV mE mv p

2 2



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Electron Diffraction-Interference

 What happens if we send electrons through a double slit apparatus?

• initially, the pattern looks random

• start to see interference

• characteristic interference pattern

Electron Diffraction-Interference

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Electron Diffraction-Interference

Louis de Broglie

(15 August 1892 – 19 March 1987)

 For his wave properties of particle, de Broglie won the Nobel Prize for Physics in 1929 for the proposal that he put forward in 1924 and proven by experiment in 1927.

 The wave-like behaviour of particles

discovered by de Broglie was used by Erwin Schrödinger in his formulation of wave mechanics.

de Broglie

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Limit of Wave Properties of Particle

 Example 1.6

 At macroscopic scale, we will not be able to see the wave nature of particles.

 A 0.1 mm diameter water droplet moving at 1 mm/s will have a de Broglie wavelength of 10^-25 m, which is tens of order of magnitude smaller than the size of a proton.

 The smallness of Planck’s constant that makes the separation of wave and particle properties so striking in the classical domain.

Evolution of the Atomic Model

 Dalton atomic model (1803)

• Elements are made of extremely small particles called atoms.

• Atoms of different elements differ in size, mass, and other properties

• The law of multiple proportions

 Thomson atomic model (plum- pudding model, 1904)

• The atom as being made up of negatively charged corpuscles orbiting in a sea of positive charge.

 Rutherford atomic model (1911)

• Atoms have their charge concentrated in a very small nucleus, and electrons are tiny particles orbiting the nucleus.

 Bohr atomic model (1913)

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Thomson Cathode Ray Experiment

 In 1897, Thomson showed that cathode rays were composed of a previously unknown negatively charged particle, and thus is credited with the discovery and identification of the electron.

 J. J. Thomson was awarded the Physics Prize in 1906 for showing that electrons are particles.

Thomson Cathode Ray Experiment

 J. J. Thomson also performed further experiments using magnetic field to determine the ratio of e/m.

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Thomson Cathode Ray Experiment

 Thomson’s method of measuring the ratio of the electron’s charge to mass was to send electrons through a region containing a magnetic field perpendicular to an electric field.

Thomson Cathode Ray Experiment

 An electron moving through the electric field is accelerated by a force:

 Electron angle of deflection:

 Then turn on the magnetic field, which deflects the electron against the electric field force.

 The magnetic field is then adjusted until the net force is zero.

 Charge to mass ratio :

y y

F ma eE

2

0 0

tan( ) v

v v v

y y

x

a t eE

    m

v0

t 

v0 0

F eEe  B

v0

E  B  v₀ E B/ ^{tan( )} ₂ ( / ) eE

m E B



 

2

tan( ) e E

m B

 

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Thomson’s Plum-Pudding model

 Thomson imagined the atom as being made up of these corpuscles orbiting in a sea of positive charge; this was his plum pudding model.

 This model was later proved incorrect when Ernest Rutherford showed that the positive charge is concentrated in the nucleus of the atom.

J.J. Thomson

Geiger-Marsden Experiment

 In 1909, Rutherford inspired Hans Geiger and Ernest Marsden to perform the gold-foil experiment.

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Geiger-Marsden Experiment

 2 sin

1

4

 N

PhET:

Rutherford Scattering

Rutherford Atomic Model

 In 1911, Rutherford put forward the theory that atoms have their charge concentrated in a very small nucleus, and electrons are tiny particles orbiting the nucleus.

Geiger Marsden Rutherford model of the atom.

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Ernst Rutherford

(30 August 1871 – 19 October 1937)

 Awarded the Nobel Prize in Chemistry in 1908 for his investigations into the disintegration of the elements, and the chemistry of radioactive substances, work that was done at McGill Univ., Canada.

 Rutherford was born in New Zealand, where he studied at Canterbury College, University of New Zealand.

 1895 Rutherford was awarded a scholarship to travel to England for postgraduate study at the Cavendish Laboratory, University of Cambridge.

 He was among the first of the 'aliens' (those without a Cambridge degree) allowed to do research at the university, under the inspiring leadership of J. J. Thomson.

Ernst Rutherford

(30 August 1871 – 19 October 1937)

 In 1898 Thomson offered Rutherford the chance of a post at McGill University in Montreal, Canada.

 In 1907 he moved to Victoria University of Manchester (today University of Manchester) in the UK.

 He is widely credited with first "splitting the atom" in 1917 in a nuclear reaction between nitrogen and alpha particles, in which

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Problem with Rutherford atomic model

 From classical EM theory, an accelerated electric charge radiates energy (electromagnetic radiation), which means total energy must decrease.

 And the radius r must decrease and the electron must fall to the nucleus!

Line Spectra

 Chemical elements were observed to produce unique wavelengths of light when burned or excited in an electrical discharge.

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Balmer Series

 In 1885, Johann Balmer found an empirical formula for the wavelength of the visible hydrogen line spectra in nm:

nm

(where k = 3,4,5…)

Johann Balmer

Rydberg Formula

 As more scientists discovered emission lines at infrared and ultraviolet wavelengths, the Balmer series equation was extended to the Rydberg equation (1888):

J. Rydberg

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Bohr’s Postulate

 In 1913, Bohr put forward 3 postulates that govern the atomic structure which can explained the spectral lines and bypass the stability problem.

 These postulates are

• An atomic system can only exist in a discrete set of stationary states, with discrete values of energy, and any change of the energy of the system, including emission and absorption of electromagnetic radiation must take place by a complete transition between two stationary states.

• The radiation absorbed or emitted during a transition between two stationary states of energies E₁ and E₂ (E₁ > E₂) is characterized by a unique frequency given by

h E E₁ ₂

 

Bohr’s Postulate

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Bohr’s Postulate

• The stationary states corresponds to a set of allowed orbits in the Rutherford atomic model. They are determined by the requirement that the kinetic energy of the electron in the orbit is related to the frequency f of the motion of the electron in that orbit by

where n = 1, 2, 3, … For circular orbits this reduces to the statement that the angular momentum takes on integer values in units of h/2p, so that

nhf v

m_e ² ₂¹

2

1 

r f v h n

n vr m L_circular _e

p

p 2

2   



 

Bohr’s Postulate

 Alternatively, for a circular orbit, we can consider that the electron is a standing wave in an orbit around the proton. This standing wave will have nodes and be an integral number of wavelengths.

p n h n r  p

2

 Thus, the angular momentum of the electron is

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Quantization of Orbital Radius

 Coulomb attraction force between the nucleus and an electron in a stationary orbit with principle quantum number n, acts as the centripetal force to keep the electron in its circular orbit

 The quantization of angular momentum gives

 Substituting the velocity above yields

 For the hydrogen atom (Z = 1),

 

n

n r

mv r

e k Ze

2

2 

p 2 nh v mr L _n  mrn

n h v 2p



 



 ² ₂ ² ₂ 4 mke

h Z

r_n n

p

B

n n a

r  ² ₂ ² ₂

4 mke a_B h

 p = 0,529  10^-10 m

Quantization of Energy

 From the force equation, we can derived the kinetic energy of an electron in a stationary orbit of principle quantum number n

 The potential energy of that electron is

 Thus the total energy of this electron is

 Substituting the expression of r_n yields

n

k r

k Ze mv

E

2 2

2 1 2

1 



n

p r

k Ze E

 2



n n

n p

k r

kZe r

kZe E

E E

2 2

2

2 1 2

1 



 













2 2 4 2 2 2 2

h mk e n E_n __Z p

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Hydrogen Energy Level

 For hydrogen atom with Z = 1,

where

2 1

1 E E_n n

2 2 4 2 1

2 h

mk E _ p e

= 13,6 eV

Hydrogen Transition Series

 From Bohr’s postulate

 Using yields

 Inserting the energy level of

n

m E

E h  

 f  c



_m _n



n m

E hc E 





1 1



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Success of Bohr Atomic Model

 It explained Hydrogen’s emission and absorption spectra.

 Only works for Hydrogen-like atom, it didn’t work for other atoms.

 With Rydberg formula given by

 Where a is called the fine structure constant

   



 



 







 2 2

2 1 1

2 1

1

m n h Z E mc

hc E^m ⁿ

n m

a



c e

0

2

4p

a 

Correspondence Principle

 The Correspondence Principle from Bohr states that the quantum theory should merge into classical theory in the limit which classical theory was known to apply.

 Consider the frequency of radiation emitted by an electron in the Bohr atomic model that jumps from the orbit with

quantum number (n + 1) to n, when n is very large.

   

 

3 2 2

2 2 2 2 2 2

2 2 2 2

1 1

1 1 2

1 1 1

2

hn Z mc

n n

h Z mc

n n h Z mc c

n n n n

a a a

 







 



 





 











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Correspondence Principle

 Classically, an electron moving in a circular orbit at radius r with velocity v will radiate with frequency of its motion

 With velocity and radius given by

 We obtain the classical frequency of radiation as

p

 p

2 1 2 

 r v

cl

mrn

n h

v 2p 



 



 ²  ₂ ² ₂ 4 mke

h Z

r_n n

p

 

3 2 2

hn Z mc

cl

  a

Genealogy

John Strutt (Rayleigh) (Phys. 1904)

J.J. Thomson

(Phys. 1906) J.C. Bose

E. Rutherford (Chem. 1908)

W. H. Bragg (Phys. 1915)

N. Bohr (Phys. 1922)

O.W. Richardson (Phys. 1928) C.T.R. Wilson

(Phys. 1927) C.G. Barkla

(Phys. 1917)

F.W. Aston

(Chem. 1922) P. Langevin