Transcript QM-lecture notes

0. Prelude -- Development of
Classical Physics and Dark Clouds
(before 20th century)
1
Classical Mechanics
Newton, Sir Isaac, PRS,
(1643 – 1727), English
physicist and
mathematician
Euler, Leonhard
(1707 -- 1783),
Swiss
mathematician.
Lagrange, Joseph Louis
(1736 -- 1813),
Italian-French mathematician,
astronomer and physicist.
Hamilton, William Rowan (1805 -1865),
Irish mathematician and
astronomer.
2
Classical Electrodynamics
Coulomb, Charles
Augustin (1736 –
1806), French
physicist
Biot, Jean Baptiste
(1774 --1862), French
Physicist;
Savart, Félix (1791 -1841),
French Physicist
Ampere, Andre
Marie (1775 -- 1836),
French Physicist
Faraday, Michael
(1791 -- 1867),
English Physicist
Lorentz, Hendrik
Antoon (1853 -1928), Dutch
Physicist
Maxwell, James Clerk (1831 – 1879), Scottish physicist
3
Classical Thermodynamics
Dalton, John (1766
-- 1844), British
chemist and
physicist.
Carnot, Nicolas
Léonard Sadi (1796
-- 1832),
French physicist.
Joule, James
Prescott (1818 -1889), British
physicist.
Helmholtz, Hermann
Ludwig Ferdinand von
(1821 -- 1894), German
physicist and physician.
Clausius, Rudolf
Julius Emanuel
(1822 -- 1888) ,
German
mathematical
physicist.
Boltzmann, Ludwig, (1844 –
1906), Austrian physicist.
Thomson, William
(Baron Kelvin)
(1824 - 1907),
British physicist
and mathematician.
Maxwell, James
Clerk (1831 –
1879), Scottish
physicist
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Classical Statistical Mechanics
Equal a priori probability postulate (Boltzmann)
Given an isolated system in equilibrium, it is found with equal
probability in each of its accessible microstates.
Canonical ensemble (isolated system)
Grandcanonical ensemble (opened system)
Boltzmann, Ludwig, (1844 –
1906), Austrian physicist.
Microcanonical ensemble
(independent system)
5
Dark Clouds
Lord and Lady Kelvin at the
coronation of King Edward
VII in 1902.
Sir William Thomson
working on a problem of
science in 1890.
William Thomson produced 70
patents in the U.K. from 1854
to 1907.
“There is nothing new to be discovered in physics now.
All that remains is more and more precise measurement.”
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Dark Clouds
"Beauty and clearness of theory... Overshadowed by two clouds..."
Nineteenth-Century Clouds over the Dynamical Theory of Heat and Light
(27th April 1900, Lord Kelvin)
Michelson, Albert
Morley, Edward
Michelson-Morley Experiment (1887)
Einstein, Albert
Planck, Max
Ultraviolet catastrophy in blackbody radiation (before October, 1900)
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I. Experiments and Ideas
Prior to Quantum Theory
(Before 1913)
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Radiation:
Blackbody Radiation and
Quanta of Energy
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Planck (1858 -- 1947), German physicist.
Planck's law of black body
radiation (1900)
Planck’s assumption (1900): radiation of a given frequency ν could only be
emitted and absorbed in “quanta” of energy E=hν
10
Radiation interaction with
matter: Photoelectric Effect
and Quanta of Light
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1. In 1839, Alexandre Edmond Becquerel
observed the photoelectric effect via an electrode
in a conductive solution exposed to light.
2. In 1873, Willoughby Smith found that selenium
is photoconductive.
3. In 1887, Heinrich Hertz made observations of the
photoelectric effect and of the production and
reception of electromagnetic (EM) waves.
4. In 1899, Joseph John Thomson (N) investigated
ultraviolet light in Crookes tubes.
5. In 1901, Nikola Tesla received the U.S. Patent
685957 (Apparatus for the Utilization of Radiant
Energy) that describes radiation charging and
discharging conductors by "radiant energy".
6. In 1902, Philipp von Lenard (N) observed the
variation in electron energy with light frequency.
1
me vk2  Ek  h  W
2
In 1905, Albert Einstein (N)
proposed the well-known
Einstein's equation for
photoelectric effect.
In 1916, Robert Andrews
Millikan (N) finished a
decade-long experiment to
confirm Einstein’s
explanation of photoelectric
effect.
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Atomic Structure
13
Nuclear atom model (1911): Ernest Rutherford
Rutherford, Ernest, 1st Baron Rutherford of Nelson,
OM, PC, FRS (1871 -- 1937), New Zealand-English
nuclear physicist.
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Classical physics: atoms should collapse!
This means
an electron
should fall
into the
nucleus.
Classical Electrodynamics: charged
particles radiate EM energy
(photons) when their velocity
vector changes (e.g. they
accelerate).
New mechanics is
needed!
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Spectroscopy
Balmer, Johann
Jakob (1825 -- 1898),
Swiss mathematician
and an honorary
physicist.
from n ≥ 3 to n = 2
Balmer series
(1885)
visible spectrum
Balmer's formula (1885)
Rydberg formula for hydrogen
(1888)
Rydberg, Johannes
Robert (1854 -- 1919),
Swedish physicist.
Rydberg formula for all
hydrogen-like atom (1888)
Bohr's formula (1913)
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II. Old Quantum Theory
(1913 -- 1924)
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Bohr's model of atomic structure, 1913
The electron's orbital angular momentum
is quantized
Bohr, Niels Henrik David (1885 -- 1962),
Danish physicist.
The theory that electrons travel in discrete orbits around the atom's nucleus, with the chemical
properties of the element being largely determined by the number of electrons in each of the outer
orbits
The idea that an electron could drop from a higher-energy orbit to a lower one, emitting a photon
(light quantum) of discrete energy (this became the basis for quantum theory).
Much work on the Copenhagen interpretation of quantum mechanics.
The principle of complementarity: that items could be separately analyzed as having several
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contradictory properties.
Bohr’s theory in 1 page
1 e2 mv 2
circular motion:

4 0 r 2
r
quantization of angular momentum: mvn rn  nKh
2
 1  me4 1
1 2
1 e
total energy: En  mvn 
 

2 2
2
2
4 0 rn
4

2
K
h
n
0


2
Quantum predictions must match classical results for large n
2
freq. of radiation at n
 1  me 4
1: h  En 1  En  
 3 2 2
 4 0  n K h
2
 1 
vn
me 4
freq. of classical circular motion:  


2 rn  4 0  n3 2 K 3h3
K  1/ (2 )
mvn rn  nh / (2 )  n
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Summary
Electron Transitions
Failures of the Bohr Model
It fails to provide any understanding of why
certain spectral lines are brighter than others.
There is no mechanism for the calculation of
transition probabilities.
The Bohr model treats the electron as if it were
a miniature planet, with definite radius and
momentum. This is in direct violation of the
uncertainty principle which dictates that
position and momentum cannot be
simultaneously determined.
The Bohr model gives us a basic conceptual
model of electrons orbits and energies. The
precise details of spectra and charge
distribution must be left to quantum mechanical
calculations, as with the Schrödinger equation.
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Prince de Broglie gets his Ph.D.
de Broglie matter wave
hypothesis (1923):
All matter has a wave-like nature
(wave-particle duality) and that the
wavelength and momentum of a
particle are related by a simple
equation.
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Davisson-Germer Experiment (1927)
Davisson, Clinton Joseph (1881 -1958), American physicist
Germer, Lester Halbert (1896 –
1971), American physicist
Electron has wave nature
(diffraction)!
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Later developments
•
•
•
•
•
•
Born’s statistical interpretation of wavefunction
Matrix mechanics (Heisenberg, Born, Jordan)
Wave mechanics (Schroedinger)
Uncertainty principle (Heisenberg)
Relativistic QM (Dirac)
Exclusion principle (Pauli)
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Birth of QM
• The necessity for quantum mechanics was thrust upon us by a
series of observations.
• The theory of QM developed over a period of 30 years, culminating
in 1925-27 with a set of postulates.
• QM cannot be deduced from pure mathematical or logical
reasoning.
• QM is not intuitive, because we don’t live in the world of electrons
and atoms.
• QM is based on observation. Like all science, it is subject to change
if inconsistencies with further observation are revealed.
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Fundamental postulates of QM
•
•
•
•
How is the physical state described?
How are physical observables represented?
What are the results of measurement?
How does the physical state evolve in time?
These postulates are fundamental, i.e., their explanation is beyond the scope
of the theory. The theory is rather concerned with the consequences of these
postulates.
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Goal of PHYS521 and 522
• We will focus on non-relativistic QM.
• Our goal is to understand the meaning of the postulates the theory is
based on, and how to operationally use the theory to calculate
properties of systems.
• The first semester will lay out the ground work and mathematical
structure, while the second will deal more with computation of real
problems.
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Linear Algebra of Quantum Mechanics
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The mathematical structure QM describes is a linear algebra of
operators acting on a vector space.
Under Dirac notation, we denote a vector using a “ket”:
v
The basic properties of vectors :
* c v is another vector
* v  w is another vector
* null vector: v  v  0
0 v 0
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A set of vectors 1 , 2 , ... , n are linear independent if
c1 1  c2 2  ...  cn n  0 implies c1  c2  ...  cn  0.
A vector space is n-demensional if
the maximum number of linearly independent vectors in the space is n.
A set of n linearly independent vectors in n-dimensional space is a basis
--- any vector can be written in a unique way as a sum over a basis:
n
v   vi i
i 1
Once the basis is chosen, a vector can be represented by a column vector:
 v1 
 
 v2 
 
 
 vn 
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A ket vector v is associated with a bra vector v in the dual space.
The inner product of two ket vectors v and w are w v  v w
*
The dual of c v is v c *.
The norm of v is defined as v 
The Schwartz Inequality: w v
2
vv .
v w .
2
2
Usually we require the basis to be orthonormal: i j  ij
A linearly independent set of basis vectors can be made orthonormal
using the Gram-Schmidt procedure.
Give an orthonormal basis
 i  , and
v   vi i ,
vi  i v
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A linear operator Aˆ takes any vector in a linear vector space to another vector
in that space: Aˆ v  v ' and satisfies: Aˆ  c v  c v   c v '  c v '
1
1
2
2
1
1
2
2
Identity operator I : I v  v for all v .
For a n-dimensional space with an orthonormal basis 1 , 2 , ..., n ,
the linear operator is completely determined by its action on the basis vectors,
and the identity operator can be express as:
n
I  i i
i 1
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Give an orthonormal basis
 i , if
Aˆ v  v ' , then vi '  Aij v j
where vi '  i v ' , vi  i v , Aij  i Aˆ j
Therefore the action of Aˆ is simply equivalent to matrix multiplication:
 v1 '   A11
  
 v2 '    A21
  
  
 vn '   An1
A12
A22
An 2
A1n  v1 
 
A2 n  v2 
 
 
Ann  vn 
and Aˆ can then be represented by an n  n matrix.
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If Aˆ v  v ' , then v '  v Aˆ † . Aˆ † is called the adjoint of Aˆ .
Give an orthonormal basis
 i , like
Aˆ , Aˆ † can also be represented by an n  n matrix
and the matrix elements of Aˆ and Aˆ † are related by
A 
†
ij
 A*ji
An operator equal to its adjoint (Aˆ  Aˆ † ), is called Hermitian.
An operator equal to minus its adjoint (Aˆ   Aˆ † ), is called anti-Hermitian.
ˆ ˆ †.
An operator is unitary if Uˆ †Uˆ  1  UU
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Unitary operator possesses the following properties:
It preserves the norm of a vector: if v '  Uˆ v , then v '  v
It preserves the inner product: if v '  Uˆ v and w '  Uˆ w , then v ' w '  v w
It transforms one orthonormal basis in the space to another orthonormal basis.
Conversely, any transformation that takes one orthonormal basis to another must be unitary.
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Eigenkets and Eigenvalues:
Aˆ ai  ai ai
Eigenvalues are roots to the characteristic polynomial
A11  
A12
A1n
A21
A22  
A2 n
An1
An 2
Ann  
det  A   I  
The set of eigenvalues of an operator satisfy:
n
n
a   A
i 1
i
i 1
n
a
i 1
i
ii
 Tr( A)
 det( A)
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Eigenkets and Eigenvalues of Hermitian Operators:
Aˆ ai  ai ai
All the eigenvalues are real.
Eigenkets belonging to different eigenvalues are orthogonal.
The complete eigenkets can form an orthonormal basis.
The operator can be written as ˆ n
A   ai ai ai .
i 1
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