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A Physics joke ….

Q: What's the difference between a quantum mechanic and an auto mechanic?

A: A quantum mechanic can get his car into the garage without opening the door!

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Postulates of Quantum Mechanics

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The position-momentum commutation relation

Further suppose that we need to calculate the product of the apple’s position and momentum. I choose to determine the product by multiplying position by momentum. You choose to multiply momentum by position. No matter, since we know that for ordinary numbers and their associated units, or xpxp xx

0 xpxp xx

Suppose that I throw an apple into the air and you photograph it as it falls to the ground using a camera fitted with a rapid autowind facility. In this way, we will be able to calculate the position and velocity of the apple at any given time.

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However, such an equation does not hold for quantum particles like electrons and photons: ixpxp xx

1iWhere:

To find the quantum mechanical equivalence, we have to assume that there must exist some function which depends on x (let us call it a wave function Ψ) on which the operators are supposed to operate. Putting this wave function into above equation gives:

ixppxxppx xxxx

)(

Choosing operator equivalent to ‘multiplication by the value of x’ and operator

to be will satisfy the above equation. Also,

x

xp

xi 2

222

. xppp xxx

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How to ‘derive’ the Schrödinger equation (again)

EVmp

2

2

Total energy of a particle

222

222

222

22222

zyx

ppp zyx

To get quantum mechanical equivalent expression, we replace the physical quantity with its quantum mechanical expression to be equal to:

2p 2

p

EVm

22

2

The result is:

This exercise demonstrates that in the equations of quantum mechanics, the values of observable quantities like position and momentum are replaced by the mathematical operators which yield these values when they operate on the wave function.

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Postulate 1: The state of a quantum mechanical system is completely described by the wave function . n

Here the subscript n serves as a shorthand for the set of one or more quantum numbers on which the wave function will depend.

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Quantum Numbers

The Bohr model was a one-dimensional model that used one quantum number to describe the distribution of electrons in the atom. The only information that was important was the size of the orbit, which was described by the n quantum number. Schrödinger's model allowed the electron to occupy three-dimensional space. It therefore required three coordinates, or three quantum numbers, to describe the orbitals in which electrons can be found.

The three coordinates that come from Schrödinger's wave equations are the principal (n), angular (l), and magnetic (ml) quantum numbers. These quantum numbers describe the size, shape, and orientation in space of the orbitals on an atom.

Dirac derived a version of the wave equation for a free electron in which space and time are treated on an equal footing. So, to completely describe a free electron, we need a fourth ‘degree of freedom’ in addition to the three degrees of freedom corresponding to translation in the x, y and z directions which requires a fourth quantum number, that is, spin quantum number (ms).

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The principal quantum number (n) describes the size of the orbital. Orbitals for which n = 2 are larger than those for which n = 1, for example. Because they have opposite electrical charges, electrons are attracted to the nucleus of the atom. Energy must therefore be absorbed to excite an electron from an orbital in which the electron is close to the nucleus (n = 1) into an orbital in which it is further from the nucleus (n = 2). The principal quantum number therefore indirectly describes the energy of an orbital.

The angular quantum number (l) describes the shape of the orbital. Orbitals have shapes that are best described as spherical (l = 0), polar (l = 1), or cloverleaf (l = 2). They can even take on more complex shapes as the value of the angular quantum number becomes larger.

In physics and chemistry, an atomic orbital is the region in which an electron may be found around a single atom.

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There is only one way in which a sphere (l = 0) can be oriented in space. Orbitals that have polar (l = 1) or cloverleaf (l = 2) shapes, however, can point in different directions. We therefore need a third quantum number, known as the magnetic quantum number (ml), to describe the orientation in space of a particular orbital. (It is called the magnetic quantum number because the effect of different orientations of orbitals was first observed in the presence of a magnetic field.)

And finally spin quantum number (ms) which identifies the "spin" or rotation of the electron about its own axis.

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Rules Governing the Allowed Combinations of Quantum Numbers

The principal quantum number (n) cannot be zero. The allowed values of n are therefore 1, 2, 3, 4, and so on.

The angular quantum number (l) can be any integer between 0 and n - 1. If n = 3, for example, l can be either 0, 1, or 2.

The magnetic quantum number (ml) can be any integer between -l and +l. If l = 2, (ml) can be either -2, -1, 0, +1, or +2.

(ms) has 2 values:+1/2 (spin up) and -1/2(spin down)

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According to Born’s interpretation, the wave functions represent probability ‘amplitudes,’ and the square of the wave function gives the probability density from the product

If the result is going to be a real quantity.

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If we wish to know the probability of finding the particle in a specific region of space, it is necessary to multiply the probability density by the volume element to which we refer. Because the ‘space’ in which the wave function is represented may be a complicated, many-dimensional configuration space, we replace the volume element by a generalized spatial element . Thus, the probability of finding the particle in the region is given by: . Because there is only one particle and it must be found some where, the integral over all spatial elements must be unity, i.e.:

dd d

d

0

1 d

This is known as the condition of normalization.

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Heisenberg is out for a drive when he's stopped by a traffic cop.

The cop says: 'Do you know how fast you were going?'

Heisenberg says: 'No, but I know where I am.'

A Physics joke ….

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Postulate 2: Observable quantities are represented by mathematical operators. These operators are chosen to be consistent with the position-momentum commutation relation.

The term observable quantity (or just ‘observable’) is used to signify all the quantities that we could, in principle, measure, such as position, momentum, energy etc.

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According to this postulate, every observable must have a corresponding operator in the theory. The operator for the total energy is called Hamiltonian operator .

H

EVm

22

2

EH

Recall:

So: Vm

VTH

22

2

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EH

The above equation has a form of characteristic of an eigenvalue equation: a mathematical operator (in this case ) operates on some function to give the same function multiplied by the value of some quantity (in this case E). Functions that satisfy such an equation are said to be eigenfunctions of the operator ( ) and corresponding values of the quantities (the energies) are called eigenvalues.

H

H

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Postulate 3. The mean value of an observable is equal to the expectation value of its corresponding operator.

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For a specific wave function :

d

dAA

nn

nnn

n

Expectation value of some operator

A

If is normalized:

Suppose that: nnn aA

Multiplying both sides of this equation from the left by and integrating over all spatial elements:

n

dadA nnnnn

nnnn adAA

n

nnn

nnn a

d

dAA

1 nn

Note that 1- when is an eigenfunction of the expectation value is exactly

2- “Mean” becomes necessary when we consider functions which are not eigenfunctions. nan

A

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If postulate 3 is to make any sense, the eigenvalue of an operator representing an observable must be a real quantity if we are to interpret it as the value of the observable, since this is something that we can, in principle, measure in the laboratory. This is an important restriction. Operators whose eigenvalue are exclusively real are called hermitian operators. Their integrals have the property that

dAdA nmmn

Where and are eigenfunctions of

An m

Furthermore, as a result of this property of the operator (hermiticity) any two eigenfunction are also orthogonal, i.e., 0 dmn

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If and normalized, so that then:

0 dmn

n m 1 dd nnmm

nmmn d

0nm when and 1 when mn mn

Eigenfunctions that are both orthogonal and normalized are said to be orthonormal.

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At the Party with the Physicists

One day, all of the world's famous physicists decided to get together for a party (ok, there were some non-physicists too who crashed the party). Fortunately, the doorman was a grad student, and able to observe some of the guests...

Everyone gravitated toward Newton, but he just kept moving around at a constant velocity and showed no reaction. Einstein thought it was a relatively good time.

Coulomb got a real charge out of the whole thing.

Thompson enjoyed the plum pudding.

Pauli came late, but was mostly excluded from things, so he split.

Hilbert was pretty spaced out for most of it.

Heisenberg may or may not have been there.

Wien radiated a colorful personality.

de Broglie mostly just stood in the corner and waved.

Stefan and Boltzman got into some hot debates.

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Suppose we want to simultaneously determine two properties, say and ,of the state described by the wave function .

nBnA

n

So, must be a simultaneous eigenfunction of both and .n

A

B

We can measure the values of two observables of some quantum state to a arbitrary precision if their corresponding operators commute. If they do not commute, they are complementary.

Consider the action of the commutator on :

],[ BA n

0

nnnnnn

nnnn

nnnn

baabBaAb

aBbA

nnnn ABBAABBABA

)(],[

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The Dirac bracket notation

mnmn

mnmn

d

AdA

nnn aA

nnn aA

≡

nn ≡ ket, ket vector, state, state vector

n≡ bra n

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Hilbert Space

The spin of an electron, as an intrinsic property of it, which does not have any counterpart in the classical world requires a spin ‘coordinate.’ This is a new, internal, coordinate of the electron which is not related in any way to the three conventional Cartesian coordinates.

Quantum particles are particles in Hilbert space. Hilbert space is an abstract, mathematical space consisting, in principle, of an infinite number of dimensions. We take as many dimensions as are needed to specify completely the state of a quantum particle.

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Expansion Theorem (Superposition Principle)

An arbitrary, well defined state vector can be expanded as a linear superposition of the complete set of eigenstates of any hermitian operator.

By ‘well defined’ we mean that the state vector has properties closely related to those of the eigenfunctions, so that it has potentially the same kind of physical interpretation, and conforms to the same set of boundary conditions.

By ‘complete’ we mean that the full set of eigenstates (also called basis set) of the hermitian operator are needed to specify completely the state .

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To make life simple, we will assume that only two eigenstates, and , are needed to specify completely the state vector , i.e. we need a basis set of only two eigenstates. The expansion theorem suggests that we mix these two eigenstates together in some proportion that has yet to be determined, so:

nm

nnmm cc

The complex conjugate of is bra, :

and are mixing coefficients which indicate how much of each eigenstate is present in the mixture. mc nc

The effect of operator on is:

A nnmm AcAcA

nnmm cc

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))(( nnmmnnmm AcAcccA

nnnmnnmnmnmmmm AcAccAccAc

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We can simply show that: 122 nm cc If is normalized.

nnmm acacA 22

(1)

Equation (1) shows that the expectation value is quite literally the ‘mean value’ of the observable: it is in fact a weighted average of the eigenvalues of the two eigenstates that make up .

A

n

m

nnmn

nmmm

a

a

AA

AA

0

0

mmm aA

nnn aA

mnnmnnnmnm AaaA

0 ( and are orthogonal.)n m

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Compton was a little scatter-brained at times. Bohr ate too much and got atomic ache. Hertz went back to the buffet table several times a minute. Faraday had quite a capacity for food. Oppenheimer got bombed. Gamow left the party early with a big bang while Hoyle stayed late in a steady state. For Schrödinger this was more a wave function rather than a social function. Born thought the probability of enjoying himself was pretty high. Shakespeare could not decide whether to be or not to be at the party. Pavlov brought his dog; which promptly chased after Schrödinger's cat. Bill Gates came to install windows. Bertrand Russell kept wondering if the cook only cooks for the guests, who cooks for the cook? Witten bought a present all tied up with superstrings. The food was beautifully laid out by Mendeleyev on the periodic table. Chadwick was handing out neutrons free of charge. Watson and Crick danced the Double Helix. Russell and Whitehead insisted on checking the bill for completeness and consistency. Gödel said it was incomplete and it can never be proved otherwise.

Descartes cogitated, 'I think I am drunk. Therefore I am at the party.'

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nnmm cc

What are the coefficients and ? Let’s answer by multiplying above equation by :

ncmc

m

mnmnmmmm ccc

Here: 1mm and 0nm Similarly: nn c

(1)

So (1) can be written as: nnmm

m nand are called inner products or projection amplitudes.

(Orthogonality)

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Euclidean space Hilbert space

ivx

jv y nnc

mmc

v

x

y

1vSuppose:

jvivv jx

coscos).( vivivx

sin)90cos().( vjvjvy

).().( vjjviiv nnmm

In general, these representations can be written:

Quantum:

Classical:

ss

s

kjis

vssv,,

).(

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Secondly, both state vectors and unit vectors have the property of orthogonality:

Quantum:

Classical:

0nm

090cos).( ji

Finally, both representations have the property of completeness:

Quantum:

Classical:

122 nm

1sincos).().( 2222 vjvi

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However, we should be under no illusions. The state vectors have properties that classical vectors can never have. The state vectors are vectors in a mathematically defined space and they can show interference effects. Whereas we can ‘measure’ vectors in classical physics, we cannot measure state vectors directly: only the modulus-squared of the state vector is accessible from experiment. The analogy between state vectors and unit vectors is a mathematical one: it offers us no help in deciding what a state vector is.