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