Ket Bra Operators

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Advanced Quantum Physics PHY604

F. A. Hashmi

Department of Physics,COMSATS Institute of Information Technology, Islamabad

Fall 2014

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Outline

1 Contents and Roadmap

2 Ket Bra Operators

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Outline

1 Contents and Roadmap

2 Ket Bra Operators

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Course

Contents

Schrodinger wave equation, bound states of simple systems, collision theory,representation and expansion theory, matrix formulation, perturbation theory,

many-body theory, second quantization,Fermi systems, Bose systems, interactionof radiation with matter, quantum theory of radiation, spontaneous emission,relativistic quantum mechanics, Dirac equation, Klein-Gordon equation, covariantperturbation theory.

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Course

Roadmap

Week 1 Quiz 0, review of quantum mechanics, bra and ket vectors, linearoperators, adjoint of a linear operator, eigenvalues andeigenvectors, completeness and expansion in eigen basis,commutativity and compatibility of observables.

Week 2 Representation theory, change of representation, coordinate andmomentum representation.

Week 3 Quiz 1, quantum dynamics, Schrodinger equation, Schrodingerequation in coordinate and momentum representation, Schrodingerequation in matrix form, Heisenberg picture, Interaction picture.

Week 4 Bound states of simple systems, particle in a box, harmonicoscillator, hydrogen-like atoms.

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Course

Roadmap

Week 5 Perturbation theory, time independent and time dependentperturbation theory, some examples of time dependentperturbation.

Week 6 Quiz 2, second quantization, Fermi and Bose systems

Week 7 Quantum theory of radiation

Week 8 Interaction of radiation with matter

Week 9 Quiz 3, scattering theory

week 10 Mid-term exam

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Course

Roadmap

Week 11 Quizes 4 and 5, Relativistic quantum mechanics, Dirac andKlein-Gordon equation, covariant perturbation theory.

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Course

Recommended books

1 Concepts in Quantum Mechanics by V. S. Mathur and S. Singh, CRC press.

2 Advanced Quantum Mechanics (4th edition) by F. Schwabl, Springer.

3 Advanced Quantum Mechanics by J. J. Sakurai, Pearson.

4 Quantum Mechanics by A. Messiah, Dover.

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Quiz 0

Answer any of the following questions

What was the need for quantum mechanics? What were theexperimental observations which could not be explained using classicalmechanics? How did quantum mechanics provide the explanations?

What is quantum mechanics? How is it different from classical

mechanics?What are the postulates of quantum mechanics?

What is wavefunction? What are the conditions on the wavefuntionof a quantum system? Give examples of some systems with their

possible wavefunctions.What is state vector? How is it different from the wavefunction? Howdoes one get information of a system from its state vector?

What are operators? What is the use and significance of Hermitianoperators in quantum mechanics?

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Quiz 0

Answer any of the following questions (continued)What is the quantum theory of measurement? What happens to aquantum system when a measurement is performed on the system?

What is the difference between expectation and eigenvalues? What isone likely to obtain if a measurement is performed on a quantumsystem? What will be the result if a large number of measurementsare performed on identical systems?

What are compatible and incompatible observables? Give someexamples of incompatible observables.Why are these incompatible?

What is energy quantization? How does it arise in quantum systems?

How does one quantum mechanically solve Hydrogen atom? What isthe solution? What are the advantages of the quantum treatmentover Bohrs atomic model?

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Operators

Operator addition

The operators can be added together. The operator addition is both commutativeand associative. IfL, M, and Nare different operators then we have

L+ M= M+L

and

L+M+N

=

L+ M

+N

i.e. the order in which different operators are added together is not important.

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Operators

Operator multiplication

The operators can be multiplied together. The operator multiplication isassociative but not commutative. IfL, M, and Nare different operators then we

have

L

MN

=

LM

N

but in general

LM= ML

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Operators

Commutator algebra

The commutator of two operators L and M is defined as

L, M

=LM ML

It satisfies following relationsL, M

=

M, L

L, M+N

=

L, M

+L,N

L, MN

=

L, M

N+ M

L,N

0 =L,M,N

+M,

N, L

+N,

L, M

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Operators

Inverse of an operator

The inverse of an operator when multiplied to the operator returns the unitoperator. i.e ifL is an operator, then its inverse operator L1 is such that we have

LL1 = I

Adjoint of an operator

The Adjoint, Hermitian adjoint, or Hermitian conjugate of an operator is the dualof that operator in bra space, and is defined by the relation

L= L

The Adjoint of a product of two operators is given by

LM

= ML

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Operators

Hermitian operator

An operator is called a Hermitian operator if it is the self adjoint i.e if we have

L

=L

thenL is a Hermitian operator. The scalar product which involves a Hermitianoperator is given by

L

=L

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Operators

Unitary operator

A unitory operator is the one which has its inverse as its Hermitian Conjugate,and so the product of the operator with its Hermitian conjugate is the identityoperator. Consider the operator U, if we have

U1 = U

then U is a unitaory operator as we have

UU = UU1 = I

A unitary operator can always be written as

U=eiL

whereL is a Hermitian operator.

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Operators

Idempotent operator

An idempotent operator is the one which satisfies

2 =

If is idempotent then so is I as

I

2=I

I

= I2 2 +2

= I

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Operators

Projection operatorAn idempotent operator that is Hermitian as well is called a projection operatorand can be used to split any ket into two orthogonal kets. If we have an arbitraryket| we can write it as

|= |+|

where we have

|= P|

|=I

P

|

and P is a projection operator. We can verify that | and | in this case areorthogonal to each other.

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Operators

Linear operator

An operatorL is a linear operator if its action on a superposition of ketsc1|1+ c2|2 (c1 and c2 being the complex numbers) is equal to the sum of itsactions on individual kets. i.e. L is linear if we have

L (c1|1+c2|2) = c1L |1+c2L |2

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Operators

Eigenkets and eignvalues

A ket which satisfies the relation

L |= |

is said to be the eigenket or eigenvector of the operator L with the eigenvalue (that in general is a complex number).

Spectrum of an operator

The set of all the eigenvalues of an operator is called the spectrum of thatoperator.

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Operators

Theorem 1

The eigenvalues of a Hermitian operator are real.

Theorem 2

The eigenkets of Hermitian operator corresponding to different eigenvalues are

orthogonal.

Expansion in eigenkets

Any arbitrary ket can be written as a linear combination of normalized eigenketsof a Hermitian operator

|=n

cn|n

The coefficient cn is given by cn =n|

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Wavefunctions

Position eigenkets

The eigenkets of position operator xare the position eigenkets denoted by |x,

and these satisfy the relation

x|x= x|x ,

where x is the position of the particle if it is in the state |x.

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Wavefunctions

Completeness of|x

The position eigenkets form a complete orthonormal basis set and any arbitraryket can be expanded in terms of position eigenkets.

|= dx|x x| .

Here the expansion coefficientx| is such that

|x||2dx

is the probability of finding the particle in a narrow strip dxaround the position x.

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Wavefunctions

Position space wavefunction

If the state of the system is represented by |, then its position spacewavefunction is given by

(x) =x| .

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Wavefunctions

Momentum eigenkets

The eigenkets of momentum operator pare the momentum eigenkets denoted by

|p, and these satisfy the relation

x|p= p|p ,

where p is the momentum of the particle if it is in the state |p.

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Wavefunctions

Completeness of|p

The momentum eigenkets form a complete orthonormal basis set and anyarbitrary ket can be expanded in terms of momentum eigenkets.

|= dp|p p| .

Here the expansion coefficientp| is such that

|p||2 dp

is the probability of finding the momentum of the particle particle in a narrowstrip dparound the value p.

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Wavefunctions

Momentum space wavefunction

If the state of the system is represented by|

, then its momentum space

wavefunction is given by

(p) =p| .

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Observables

PostulateWith every dynamical variable/observable is associated a linear Hermitianoperator.

Examples

observable operatorposition coordinate x xposition vectorr rxcomponent of momentum px i

x

momentump ikinetic energy p2/2m (h2/2m)2potential energy V(r, t) V(r, t)total energy (p2/2m) +V(r, t) H=(h2/2m)2 +V(r, t)

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Observables

Compatible observables

The observables whose operators commute with each other.

Can be measured simultaneously with arbitrary precision.

Have a common set of eigenvectors.

Examples

any component of the angular momentum and its modulus square.the x coordinate of the position of a particle and the y component of

its momentum.

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Observables

Incompatible observables

The observables whose operators do not commute with each other.

Can not be measured simultaneously with arbitrary precision.

measurement of one observable renders the earlier measurement of theother observable obsolete.

Do not have a common set of eigenvectors.

Examples

any two components of the angular momentum of a particle.the x coordinate of the position of a particle and the x component ofits momentum.

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