Infinite square well (particle in a box) Announcements: • Homework set 8 is posted and is due next Wednesday, Oct. 30.
• Still have homework to return
Today we will be investigating the infinite square well (particle in a box). http://www.colorado.edu/physics/phys2170/
Physics 2170 – Fall 2013
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Identities eikx= cos kx + i sin kx e-ikx= cos kx - i sin kx = 1/eikx cos kx = (eikx + e-ikx)/2 sin kx = (eikx - e-ikx)/2i
http://www.colorado.edu/physics/phys2170/
Physics 2170 – Fall 2013
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Interesting Facts for Friday Chewing gum was invented by a dentist, named William Semple - as a way to exercise your jaws. First person to patent chewing gum in 1869. Adams patented chicle gum that people now spend 2 billion dollars a year on. Was originally made by tapping trees, now done chemically. A solar panel 100 miles by 100 miles (161x161km) in the Mojave Desert (USA) could replace all the coal now burned to generate electricity in the entire U.S.
http://www.colorado.edu/physics/phys2170/
Physics 2170 – Fall 2013
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Interesting Friday Facts cont. Lake Baikal in Russia contains more water than all the North American Great Lakes combined. One fifth of all fresh water in the world is contained in Lake Baikal. It is 400 miles long!
http://www.colorado.edu/physics/phys2170/
Physics 2170 – Fall 2013
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More facts cont. Mexico City is sinking at a rate of 18 inches (46cm) per year as a result of draining water. The first ten feet of the ocean hold as much heat as the Earth's entire atmosphere. Hawaii is moving toward Japan at 10 centimeters (4 inches) every year. The temperature in Fahrenheit can be determined by counting the number of cricket chirps in 14 seconds and adding 40. Comes from the Farmers Almanac http://www.colorado.edu/physics/phys2170/
Physics 2170 – Fall 2013
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Cricket Chirps – study performed in Boulder 2007 Education Study
http://www.colorado.edu/physics/phys2170/
Physics 2170 – Fall 2013
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Review of Schrodinger Eq. 1. It must be consistent with a. λ = h/p b. v = E/h c. E = p2/2m + V (note non-relativistic)
2. It must be linear in Ψ(x,t). A linear combination of waves can be solutions. Ψ(x,t) = a1Ψ1(x,t) + a2Ψ2(x,t) 3. The potential energy V is, in general, a function of x and t. So far have looked at problems where V is a constant or a function of x. (ie V = V(x) )
http://www.colorado.edu/physics/phys2170/
Physics 2170 – Fall 2013
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Free particle The general (spatial part) solution for a free particle is: Using some identities you will be proving in the homework, this can also be written as To get the full wave function we multiply by the time dependence:
This is the sum of two traveling waves, each with momentum ħk and energy ħω. The C wave is moving right and the D wave is moving left. Lets pick the right going wave and check it one more time. http://www.colorado.edu/physics/phys2170/
Physics 2170 – Fall 2013
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Check free particle solution in TDSE TDSE: For a free particle, V(x,t)=0. The solution for a particle moving to the right is If we plug this into the TDSE, what do we get?
Putting it all together gives:
It all works! http://www.colorado.edu/physics/phys2170/
Kinetic energy Physics 2170 – Fall 2013
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Total energy 9
Clicker question 1
Set frequency to AD
Left moving free particle wave function is: Can also write as The probability of finding the particle at x=0 is _________ to the probability of finding the particle at x=L. A. Always larger x=0 B. Always smaller C. Always equal D. Oscillates between smaller & larger E. Depends on other quantities like k
x=L
or http://www.colorado.edu/physics/phys2170/
Physics 2170 – Fall 2013
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Clicker question 1
Set frequency to AD
Left moving free particle wave function is: Can also write as The probability of finding the particle at x=0 is _________ to the probability of finding the particle at x=L. A. Always larger x=0 B. Always smaller C. Always equal D. Oscillates between smaller & larger E. Depends on other quantities like k
or http://www.colorado.edu/physics/phys2170/
x=L Get probability from |Ψ(x,t)|2.
Probability is constant (same everywhere and everywhen) Physics 2170 – Fall 2013
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Electrons in wire We just solved the free particle in 1D problem. Remember, free particle means there is no force acting on it. An electron in a very long (technically infinite) copper wire with no voltage on it would have the same wave function.
Next: Wave function of electron in a short (length a) copper wire. 0
a
Use TISE:
We will first figure out V(x), then find the solution, and finally analyze what the solution means physically. http://www.colorado.edu/physics/phys2170/
Physics 2170 – Fall 2013
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Clicker question 2
Set frequency to AD
What can we say about V(x) for an electron inside a copper wire which extends from x=0 to x=a? A. V(x) is smaller in the region 0
