15minute.course.in qm

8/13/2019 15minute.course.in QM

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One assumes:

(1) energy, E (-

/i)/t(2) momentum, P (/i)(3) particle probability density, (r,t)

= i/x+ j/y+ k/z = 2 = 2/x+ 2/y2+ 2/z2

Quantum Mechanics is a

Mathematical Model

These can not be derived

-- they are postulates!t = time

The gradient operator

Plancks constant/210-34Joule-sec


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In classical physics we write kinetic energy as

T =mv2 = (mv)2/2m = P2/2m

Using P (/i) (QM assumption)(1/2m)PP

(1/2m)(

/i)

2T (-2/2m)2

deriving the Schrodinger Equation

P= mv= momentum


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E = kinetic energy + potential energy

(-/i)/t = (-2/2m)2 - e2/r

Schrodinger Equation forHydrogen atom

Coulombpotential energy

Finally, these operators act on (r,t)!


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Schrodinger Equation for H atom

The physics in this equation is not easily visualized.

*(r,t) (r,t) dV representsthe probability that the electron in the hydrogen atomcan be found within a volume, dV= dxdydz, at (r,t).

Since the electron must be somewhere,*dxdydz= 1

This turns out to be veryimportant!

[(-2

2

/2m)2 - e 2/r ] (r,t) = i/t(r,t)


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Model predicts experimental atomic structure

observed in the laboratory (model is correct!)

Model implies that an electron behaves like awave when it is confined to 10-8cm distances.

All one needs is differential equations to solve

for (r)!

HydrogenSchrodinger Equation time dependence

[(-2

2 /2m)2 - e 2/r ] (r)e iEt/= E (r) e iEt/ time dependence is exponential, E= constant

(-/i)/te iEt/ = Ee iEt/


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A strange outcome is that the mathematical boundary

condition on ,*dxdydz = 1

limits the allowed values for E.

Quantization of Energy!

Quantization is a

mathematical result!

[(-2

2 /2m)2 - e 2/r ] E(r) = E E(r) its an eigenvalue equation!

Ensuring that the

integral does not

diverge is not easy!

With the time dependence factored out


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Old classical model for the hydrogen atom

The mathematics helps us describe

and quantify this electron cloud.


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Details!

All this comes from

requiring that the

Integral converge.


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Everything that we know is made of atoms: the

mathematics that determines the structure of atoms

and the molecules formed from them is crucial to allof chemistry, biology and materials science!

ICE


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Graphene


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Cosmology and such

Those who study the smallest particleselectrons,quarks, neutrinos and the rest of the basic building

blocks of our world have an extraordinary story --

about how our universe developed from the big bang.

The story is based on mathematics.


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First of all, they start with a kind of Schrodinger equation

model one which is appropriate for particles which are

moving very fast. It is called QuantumField Theory. In this

field theory there is a field operator, (r,t), for the electron.

Then they do something rather startling: they postulate that thelaws of physics (their equations) should be invariant under a kind

of rotation,called a gauge transformation.

They discovered that it wont work unless they have a photon(called a gauge particle) to undothe rotation.


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AA

invariance

Note that the photon field

must also be transformed.

1. Initial state 2. Rotate

3. Transform A 4. Final state The photon

undoes the

rotation and

preserves the

symmetry!


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Thank you!


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From this field operator a kind energy operator, called a Lagrangian density,L, is constructed. A similar operator in classical physics is known to reproduce

Newtons law, F= ma, when one minimizes Ldt over the path of the motion.Then the particle physicists do a very interesting thing:

They demand that this Lagrangian energy operator be invariant under

a special mathematical operation, called a gauge transformation.

The gauge transformation is a kind of rotation which changes the complex

phase of the electron by an arbitrary function which depends on where

(in space-time) the particle is.


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Cosmology and such

Those who study the smallest particleselectrons, quarks, neutrinos and

the rest of the basic building blocks of our world have an extraordinary

story -- about how our universe developed from the big bang.

The story is based on mathematics.

First of all, they start with a kind of Schrodinger equation modelone which

is appropriate for particles which are moving very fast. It is called Quantum

Field Theory. In this field theory there is a field operator, (r,t), for the

electron. The field

operator is a linear combination of all possible free particle states for anelectron. With this field operator one can create (or destroy) a free electron

at any point in space, and with any energyif the math calls for it.


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This gave them the idea to look at the expanding, cooling universe as a

series of symmetries which are broken as particle fields Crystallize out.

It has been so successful that it is now the accepted story of how the

universe started from a small dense, hot system to the vast array of

galaxies and stars as we know it today.

What they find is that the invariance can not occur unless one introduces

another field, called the gauge field. This gauge field emerges as the force

field by which electrons interact with each other: the photon field!

In other word, the gauge SYMMETRY demands that the photon exists!

Furthermore, the modified Lagrangian (wth the photon field) prescribes

exactly how the electron interacts with the photon!


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= 2, m = -2,-1, 0, 1, 2

15minute.course.in qm

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