there will be written exam and 30% of points are for...

Anna Lipniacka www.ift.uib.no/~lipniack/Introduction

Anna Lipniacka, room 303, 82803, www.ift.uib.no/~lipniack/ Your teacher for for “Kvantemekanikk” I am a particle physicist with mostly CERN activities. For this and other courses I am/was teaching look at my web page. There will be written exam and 30% of points are for homeworks problems

Lectures Thursday 9:15 12:00 room 368 Tutorials ( exercises): Tuesday 12:1514:00 room 368 Consultations: I am happy to fix the time if you need it. For example Wednesday 15:0016:00

http://www.ift.uib.no/~lipniack


Technical details about the Course:

web page www.ift.uib.no/~lipniack/phys201_v10/ course notes and exercises are published on the web page exercises are very important. The exercise problems will be published on the WEB on Sunday the week before so you have time to think about them.Course book:

P.C.Hemmer: Kvantemekanikk, 2. utgave (Tapir 2000)Kap. 1 er kursorisk. Bør leses som en oppfrisking og utdyping av PHYS 115Final Curriculum:-----------------------------------------------------Kap 2.1-2.3,2.4, 2.5, 2.6.1, 3.1, 3.4, 3.5, 4.1-4.6, 5.4, 5.6-5.9, 6.1-6.5,(7.1 without 7.1.3), 7.2, 7.3, 8.1, 8.2, 8.3, 8.4, 11, 13.1, 13.2,13.3, 13.4, (13.5 without 13.5.5-13.5.6)----------------------------------------------------



Quantum physics in this room

1) the existence of atoms2) chemical reactions.3) light spectrum from the SUN and other “hot bodies”.if atoms in hot bodies were classical oscillators (electrons on a spring) most of energy emitted due to thermal atomic oscillations would be at very high frequency the radiation would simply kill us.

I , T =2

2

c2kT J T =∫

0

∞

I , T d =∞

I ,T =22

c2

he h/ kT−1

Planck formula

note E=h , I ,T =2

2

c2

E

eE/ kT−1

≃2

2

c2 E e−E /kT

for E ∞



Most striking facts about quantum world we can predict only probabilities of certain events happening certain variables can have only discrete values and change only by a discrete amount. Angular momentum and spin are variables which are always quantized : it can only be :

as angular momentum (or spin) is a variable of dimension: momentum*distance, it strikes us that there must be a limit on “granularity”

all physical states behave like waves and particles at the same time: waves: like for waves their “position” is described by a probability amplitudes which can interfere in some situations particles: certain variables in certain situations appear in “quanta”

0,12ℏ ,ℏ ,

32ℏ ,2ℏ ...

And so on.

p x≥12ℏ E t≥

12ℏ

Most striking: everything which is not forbidden by conservationlaws can happen ! Vacuum is the world most complicated structure!



Light as particles and wavesInterference : wave property of lightphotoelectric effect: particle property of light

Number of photonsregistered proportionalto intensity of lightgiven by wave interferenceand diffraction pattern



This Course: Deals with “formal” nonrelativistic quantum mechanicswe will be dealing mostly with “tool box”= formalism,we will not discuss very many applications, only simpleones.

Learning the QM “tools” is essential to understand all modern physicsand most of modern technology in “nano”, “bio” , medical etc.

nonrelativistic QM,Schroedinger equation

atomic physics

“multi body “ approximate methods

molecular physicssolid state physicsbio applicationsrelativistic, dirac

kleingordon equations..

statistical physics

(gases, liquids, solid, plasmas,nuclear state, stars)

nuclear physics quantum fieldtheory particle physics, cosmology



“Classical and quantum” measurements and probability distributions

1) We learn about real world by making measurements (observations) andconstructing models of reality. We compare measurements with model predictionsWe compare measurements with model predictions2) Most of “everyday” measurements and observations concern macroscopicobjects. Examples: a hour of arrival of the bus, number of people on a ferry BergenHansholm who are drunk on a given day, time it takes for a teacandleto burn to the end, the length of your shoe, the salinity of water from the tap inyour home etc etc.

The fact of life is the following:If you measure the same* variable on the same* objectin the same* condition you are likely get different resultseach time. If the object is “classical” this is due to the factthat the same* conditions can only be reproduced with a givenaccuracy. If the object is “quantum” this is inherently due tothe quantum nature.

What to do ? The upside of the story is: the probability distribution of gettingWhat to do ? The upside of the story is: the probability distribution of gettinga given result can be measured/calculated, and the “average” of manya given result can be measured/calculated, and the “average” of manymeasurements of the same* object can be predicted/measured/calculatedmeasurements of the same* object can be predicted/measured/calculated/compared with the model./compared with the model.

(same* = as identical as you can get)



Probability and probability density functions, averages,standard deviations, expectation values

Let (v) be some variable we can measure. Let p(v,a) be a probability densitygiven some condition (a). For example (a=”no rain today”) This means : for continuous (v) probability that v is between v and v+dv for discrete (v) probability that v takes a given valuep(v,a) should be normalized to make sense :p(v,a) should be normalized to make sense :

pv ,adv

pv ,a

∫−∞

∞

pv , adv=1

If we make many measurements of v and take the average we should get:

=∫−∞

∞

v pv dv

v

p(v)



Probability and probability density functions, averages,standard deviations, expectation values

=∫−∞

∞

v pv dv

v

p(v)

is the expectation value of v = E[v] , called mean value

In general we can define an expectation value of any function of v,say u(v)

one useful expectation value is variance V, this is an expectation valueof the squared distance from the mean :

E [u v ]=∫−∞

∞

uv pv dv

2≝V [v ]≝E [v−

2]=∫

−∞

∞

v−2 p vdv=∫

−∞

∞

v2−2 v

2 p v dv=E [v2

]−2



Some well known probability distributions, here denoted f().

Binomial: Discrete. Let n will be a number of hits of the small target, out of N throws to the whole target. p

1p

f n , N , p =N !

n! N−n!pn 1−pN−n

Mean =N*p, variance = N*p*(1p)

Now, what is the limit of this function if:

we get Poissonian distribution, which has both variance and mean equal to xample: Lots of atoms, each of them has a small probability of decayingin a given time, we know that on average atoms decay. We get the probabilityof “n” atoms decaying from the above function.

N∞ , p 0, N∗p=const=

f n , N , p =N !

n! N−n!pn 1−pN−n f n ,=

n e−

n !

Gaussian: f x , ,=1

2e−x−

2

22



Example:Quantum statistics:entropy, temperature

Entropy of a system is DEFINED as : σ = ln ( g )g is the number of states of the system (here two boxes with arrows, arrow can beup or down)

logarithm is used as a matter of convenience, as g is typically very big ~ 1020

We see this variable has right properties. Say we want to find the total number of states for a combination of two DISCONNECTED systems 1 and 2

1 2

g =1 2

gg *

=1 2Now if we CONNECT the systems 1 and 2, the number of states can stay the same or increase so

12≥1 2



Fundamental Temperature

Lets consider two isolated systems, each characterized by energy (U1', U2'), and number of particles N1, N2.

What happens if the systems are brought in contact so the ENERGY can flow in between them (but not the particles) => THERMAL CONTACT

U1, N1 U2,N2

U1+U2=U, dU1+dU2=0

U1', N1 U2',N2

U=U1'+U2'

g1(U1',N1) g2(U2',N2)

gN 1N2 ,U =g N ,U =∑U 1

g1N1 ,U 1g2N2 ,U−U1

What will be the final U1 and U2 ? The energy should flow in such a way to allow that the combined system can occupy maximal number of states> more probable state. Assume there exist U1 maximizing g= g1*g2, so for small changes we should have :

dg=∂ g1

∂U 1

N1

g2dU 1∂ g2

∂U 2

N 2

g1d U 2=0, dU 1dU 2=0

1g1

∂ g1

∂U 1

N 1

=1g2

∂ g2

∂U 2

N2

∂ ln g1

∂U 1

N1

=∂ ln g2

∂U 2

N 2

∂1

∂U 1

N1

=∂ 2

∂U 2

N2

The total number of states the combined systemcan take is the sum over all possible energy shearing



Famous Boltzmann factorexample

U

U=0

0

U ε

U=ε

0

Imagine a small system, with only two states allowed, one ofenergy=0 and one of energy =ε . The system is in thermal contactwith big reservoir, and the total energy of both systems = U0

g( U ε )0

g( U )0 What is the probability P(ε ) that the small system will be in the energy

state ε ??

Fundamental assumptions,probability~~number of states:

PP 0

=g U 0−

gU 0≝

exp[ U 0−]

exp [ U 0]

number entropy of states

U 0−≃ U0−∂

∂U 0

N

= U 0−

exp[ U 0−]≃exp[U 0]exp −

PP 0

=exp −

leading to Planck formula

energy/temperature

P 1

P 2=

exp−1

exp−2



Curriculum & main themes:

Chap. 2 (2.1-2.3,2.4 2.5, 2.6.1)Basic principles, Operators & Functions and their propertiesThe Schroedigner equation (separation, probability currents)

Chap. 3 (3.1, 3.4, 3.5)Delta functions & The Harmonic Oscillator -I

Chap. 4 (4.1-4.5, 4.6)Commutation relations & implications, Parity, Ehrenfest & Uncertainity principle

Chap. 5 (5.4, 5.6, 5.7-5.9)Angular momentum & Central (=spherical) symmetry: The Coulomb potential & hydrogen atoms

This course :



Chap. 6 (6.1-6.5)

Bra-Ket formalism & operators, vectors & matrix representation. Harmonic Oscillator - II

Chap. 8 (8.1 -8.4)

General Spin with Bra-Ket formalism, Representation of small spins (1/2, 1), Addition of spins, identical particles

Chap. 7 (7.1 without 7.1.3, 7.2, 7.3)

Time independent perturbation theory & Variational Method

Chap. 11

Time dependent basis solution, perturbation theory & the golden rule

Chap. 13 (13.1 - 13.5)

Classical scattering & basic concepts. Quantum mechanical time independent potential scattering problem, the scattering amplitude & the Born approximation (use it / not derive it)



i h∂r , t

∂ t= Hr , t =[

p2

2mV r ]r , t

i h∂r , t

∂ t=[−

h2

2m∇2V r ]r , t

Schrodinger equation

inserting momentum operator

Er =[−h2

2m∇

2V r ]r

stationary solutions



atomic physics

timedependent Schrodinger equation,atomradiation interactions quantum dotsnanotech ( Dorte Madsen , Bodil Holst)

Physics

Ladislav KocbachJan Petter Hansen

Nuclear physics

Experimental search for QuarkGluon Plasmaat CERN and Brookhaven experimentsApparatus (detector) constructions (microelectronics)theoretical study of quarkgluon plasmatheoretical study of extreme nuclear states

Dieter RohrichJoakim Nystrand Microelectronics Laszlo CsernaiJan Vaagen

Particle Physics

Participation in ATLAS experiment at CERNcosmic muon detector in IFT.Study of Dark Matter detection at ATLAS,search for Higgs particletheoretical study of Higgs models,“dark matter” related theories, neutrinos

GRID computing (ATLAS, ALICE)

Bjarne Stugu,

Anna LipniackaGerald EigenPer Osland

Anna LipniackaHaavard Helstrup

space physics solar and atmospheric physicsINTEGRAL satellite, apparatus construction(gamma rays from space)

Johan Stadnes, Kjartan OlafssonNikolai Ostgard

http://www.ift.uib.no/partikkelfysikk/



Numerous exchanges for undergraduate and graduate studentsthe Department is involved in:

NordPlus ( Nordic Countries), Aarhus(atomic physics)

Erasmus ( Europe)Heidelberg ( Dieter Rohrich)Freiburg, Sheffield ( Anna Lipniacka)(possibly also Durham and Udine )Graduate School ( HeidelbergBergen, Dieter Rohrich)CERN summer school, CERN technical master programme: many other schools


there will be written exam and 30% of points are for...

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