applying hyperbolic functions to quantum tunneling and ... · how they help with radiation problems...

UCF EXCEL

I. Warmup questions concerning calculusII. Hyperbolic functions cosh(x) and

sinh(x)III. Excellent properties of the derivative is

cosh(x) and sinh(x)IV. How they help with radiation problems

Applying hyperbolic functions to quantum tunneling and electromagnetic wave problems in physics and

engineering (Lecture 1)

Dr. Thomas J. Brueckner, Department of Physics

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1. If 10a = b, then log10(b) =

A. log10(b) = 0.1a

B. log10(2b) = 2a

C. log10(b) = a

D. log10(b/100) =100+a

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Let’s take it apart!

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Let’s take it apart!

So, you need a w value of 1.0955!

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Let’s pull this apart!

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What it means geometrically

Left of origin, in the

negatives…

…it is up side down!

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Differentiation of a sum = sum of derivatives

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Differentiation of a sum = sum of derivatives

YAY!

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Why does this catch the eye of a mathematician or a scientist?

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A. Differentiate once, you get the other guy.

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A. Differentiate once, you get the other guy.

B. Differentiate again, you get back home!

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C. Very useful!!!D. Because

derivatives have various physical interpretations, like vectors!

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What our goal is this afternoon

Penetrate material with electromagnetic waves that reshape into hyperbolic functions.

2L

y

x z

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What our goal is this afternoon

Penetrate material with electromagnetic waves that reshape into hyperbolic functions.

2L

y

x zCh. 2 in

Applicationsof

Calculus II

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What are we talking about?

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cosh

sinh

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weird…but handy in physics!

cosh

sinh

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Recall: Euler’s formula…

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Recall: Euler’s formula…

We need this today!

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How are they proportional?

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What we are talking about

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cos(kx), sin(kx)

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cos(kx), sin(kx)

exp(kx), exp(-kx)

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cos(kx), sin(kx)

cosh(kx), sinh(kx)

exp(kx), exp(-kx)

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k = 2.7 for each

cos(kx), sin(kx)

cosh(kx), sinh(kx)

exp(kx), exp(-kx)

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Each graphed over [-1, 2]

k = 2.7 for each

cos(kx), sin(kx)

cosh(kx), sinh(kx)

exp(kx), exp(-kx)

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Here we go. Let’s look at derivatives.

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Let’s check sines, too!

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In general: after two diff’s you get back to same function… but with an extra factor of -ω2.

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General derivatives relationships here:

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What do these two add up to?

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0

What do these two add up to?

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We like that because…

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A. That kind of derivative relationship is needed in spring systems, e.g.,

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A. That kind of derivative relationship is needed in spring systems, e.g., Oscillators

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A. That kind of derivative relationship is needed in spring systems, e.g., Oscillators Vibrating systems

Simple seismometer

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B. F = -kx in springs

Simple seismometer

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B. F = -kx in springsC. F = ma…as always

Simple seismometer

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B. F = -kx in springsC. F = ma…as alwaysD. a = second deriv. of x!

Simple seismometer

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Brainiac iClicker question coming…

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For this derivatives relationship, what will the ω2 factor be?

A. It comes out to be k/m.

And the temporal scale in ω depends upon the mass (m) of the object on the spring and on the strength (k) of the spring.

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Onward to radiation

in vacuum E” = 1

c2

d2

dt2 E

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Onward to radiation

A. The relationship for radiation involves the electric field (E) and magnetic field (B)in vacuum

E” = 1

c2

d2

dt2 E

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Onward to radiation

A. The relationship for radiation involves the electric field (E) and magnetic field (B)

B. They are coupledin vacuum

E” = 1

c2

d2

dt2 E

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Onward to radiation


B. They are coupledC. In space-time

in vacuum E” = 1

c2

d2

dt2 E

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Onward to radiation


B. They are coupledC. In space-timeD. With second

derivative in space and in time.

in vacuum E” = 1

c2

d2

dt2 E

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Onward to radiation


B. They are coupledC. In space-timeD. With second

derivative in space and in time.

E. Wiggles that move

in vacuum E” = 1

c2

d2

dt2 E

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Spacetime implications

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6. Sines and cosines of time and space

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7. Sin(kx ± ωt)

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7. Sin(kx ± ωt)8. Cos(kx ± ωt)

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7. Sin(kx ± ωt)8. Cos(kx ± ωt)9. Kosher ω and k,

if…

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7. Sin(kx ± ωt)8. Cos(kx ± ωt)9. Kosher ω and k,

if…

d2

dt2 E = −ω2E

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Another brainiac question coming…

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Space-time dependence of radiation and its speed of propagation

Speed of light couples the temporal and spatial dependences…

a.k.a. frequency and wavelength!

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What is this physical relationship to c?

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A. Dispersion relation (fancy terminology)

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B. Refraction of light (transmission)

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B. Refraction of light (transmission) E.g., prisms disperse

the colors of sunlight

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the colors of sunlightC. Absorption of light

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the colors of sunlightC. Absorption of light

E.g., Superman, lead absorbs xrays.

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Penetrating a material with radiation

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A. Inside material, the physics changes.

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A. Inside material, the physics changes.B. Light moves more slowly.

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A. Inside material, the physics changes.B. Light moves more slowly.C. Energy is absorbed from E and B.

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A. Inside material, the physics changes.B. Light moves more slowly.C. Energy is absorbed from E and B.D. Heat flows, outer surfaces cool off

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A. Inside material, the physics changes.B. Light moves more slowly.C. Energy is absorbed from E and B.D. Heat flows, outer surfaces cool offE. New spatial and temporal derivatives

for E and B fields.

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A. Inside material, the physics changes.B. Light moves more slowly.C. Energy is absorbed from E and B.D. Heat flows, outer surfaces cool offE. New spatial and temporal derivatives

for E and B fields.» sine, cosine = NO GO!

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Now a new derivatives relation must hold.

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F. The old derivatives relationship changes, i.e.., for Ez...

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F. The old derivatives relationship changes, i.e.., for Ez...

G. New deriv. rel. means new functions do the work.

H. Hyperbolic functions, cosh(u) and sinh(u).

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Next Monday:

A. Radiation penetrating “cheese”

B. Quantum tunnelingC. Space-time structure

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applying hyperbolic functions to quantum tunneling and ... · how they help with radiation problems...

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