http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 1
Getting to the Schrödinger equation
• Homework #6 is available to be picked up.
Announcements:
Erwin Schrödinger (1887 – 1961)
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Uncertainty Example from Monday ….. • An engineer is dropping marbles from a ladder of
height H with sophisticated equipment and trying to hit a crack on the floor everytime. Despite his great care, show that she will miss the crack by an average distance of where g is the acceleration due to gravity.
Physics 2170 – Fall 2013 2
€
/m( )1/ 2 H /g( )1/ 4
H = ½ a t 2
€
t =2Hg ΔE Δt ≥ ħ/2 ΔΕ ≥ħ/(2Δt)
ΔΕ =(Δp)2/2m
€
Δp = 2mΔE Δx Δp ≥ ħ/2
€
Δx ≥ 2Δp
=2
12mΔE
=2
Δtm
=12m⎛
⎝ ⎜
⎞
⎠ ⎟ 1/ 2
Δt( )1/ 2 =12m⎛
⎝ ⎜
⎞
⎠ ⎟ 1/ 2 H
g⎛
⎝ ⎜
⎞
⎠ ⎟
1/ 4
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Wave Function Comments • Wave Function is complex – means can have
imaginary and real components – doesn’t have to have both.
• The fact that wave functions are complex makes it obvious we should not attribute to wave functions a physical existence (ie water waves). Shouldn’t try to answer exactly what is waving and what is it waving in!
• We shall find that wave functions contain all the information which the uncertainty principle allows us to know about the associated particle.
Physics 2170 – Fall 2013 3
http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 4
Classical waves obey the wave equation:
Where we go from here We will finish up classical waves
Then we will go back to matter waves which obey a different wave equation called the time dependent Schrödinger equation:
On Wednesday we will derive the time independent Schrödinger equation:
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A. Standing waves have points called antinodes which are motionless
B. Standing waves have points called nodes which are motionless
C. Standing waves can be constructed from two traveling waves moving in opposite directions
D. A and C are both true E. B and C are both true
Reading quiz 1 Set frequency to AD Which of the following is a true statement about standing waves?
http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 6
A. Standing waves have points called antinodes which are motionless
B. Standing waves have points called nodes which are motionless
C. Standing waves can be constructed from two traveling waves moving in opposite directions
D. A and C are both true E. B and C are both true
Reading quiz 1 Set frequency to AD Which of the following is a true statement about standing waves?
Antinodes move the most while nodes do not move at all.
http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 7
Vibrations on a string: Electromagnetic waves:
Example Wave Equations You Have Seen
v=speed of wave c = speed of light
x
y E
x
Magnitude is non-spatial: = Strength of Electric field
Magnitude is spatial: = Vertical displacement of String
Solutions: E(x,t) Solutions: y(x,t)
http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 8
Solving the standard wave equation
1. Guess the functional form(s) of the solution 2. Plug into differential equation to check for correctness, find
any constraints on constants 3. Need as many independent functions as there are derivatives. 4. Apply all boundary conditions (more constraints on constants)
The standard wave equation is
Generic prescription for solving differential equations in physics:
http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 9
A. B. C. D. E. More than one of the above
Clicker question 2 Set frequency to AD
The standard wave equation is
Step 1: Guessing the functional form of the solution
Which of the following function forms is a possible solution to this differential equation?
http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 10
A. B. C. D. E. More than one of the above
Clicker question 2 Set frequency to AD
The standard wave equation is
Step 1: Guessing the functional form of the solution
Which of the following function forms is a possible solution to this differential equation?
The A. form leads to or
Different functions on the left side and right side. This is incorrect.
The B. form leads to which doesn’t work.
The C. form leads to which doesn’t work.
http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 11
Claim that is a solution to
Step 2: Check solution and find constraints
Time to check the solution and see what constraints we have
LHS:
RHS:
Setting LHS = RHS:
This works as long as
We normally write this as
so this constraint just means or
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and we have the constraint that
Since the wave equation has two derivatives, there must be two independent functional forms.
Constructing general solution from independent functions
The general solution is
Can also be written as
x
y t=0
We have finished steps 1, 2, & 3 of solving the differential equation.
Last step is applying boundary conditions. This is the part that actually depends on the details of the problem.
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Boundary conditions for guitar string
0 L
Guitar string is fixed at x=0 and x=L.
Wave equation
Functional form:
Boundary conditions are that y(x,t)=0 at x=0 and x=L.
Requiring y=0 when x=0 means which is
This only works if B=0. So this means
http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 14
Clicker question 3 Set frequency to AD Boundary conditions require y(x,t)=0 at x=0 & x=L. We found for y(x,t)=0 at x=0 we need B=0 so our solution is . By evaluating y(x,t) at x=L, derive the possible values for k.
A. k can have any value B. π/(2L), π/L, 3π/(2L), 2π/L … C. π/L D. π/L, 2π/L, 3π/L, 4π/L … E. 2L, 2L/2, 2L/3, 2L/4, ….
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Clicker question 3 Set frequency to AD Boundary conditions require y(x,t)=0 at x=0 & x=L. We found for y(x,t)=0 at x=0 we need B=0 so our solution is . By evaluating y(x,t) at x=L, derive the possible values for k.
A. k can have any value B. π/(2L), π/L, 3π/(2L), 2π/L … C. π/L D. π/L, 2π/L, 3π/L, 4π/L … E. 2L, 2L/2, 2L/3, 2L/4, …. To have y(x,t) = 0 at x = L we need
This means that we need
This is true for kL = nπ. That is,
n=1
n=2
n=3
So the boundary conditions quantize k. This also quantizes ω because of the other constraint we have:
http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 16
Summary of our wave equation solution 1. Found the general solution to the wave equation
€
y = Asin(kx)cos(ωt) + Bcos(kx)sin(ωt)or
2. Put solution into wave equation to get constraint
3. Have two independent functional forms for two derivatives
4. Applied boundary conditions for guitar string. y(x,t) = 0 at x=0 and x=L. Found that B=0 and k=nπ/L.
Our final result:
€
y = Asin(kx)cos(ωt)with and
n=1
n=2
n=3
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Standing waves
Standing wave
Standing wave constructed from two traveling waves moving in opposite directions
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Examples of standing waves
Same is true for electromagnetic waves in a microwave oven:
For standing waves on violin string, only certain values of k and ω are allowed due to boundary conditions (location of nodes).
We also get only certain waves for electrons in an atom. We will find that this is due to boundary conditions applied to solutions of Schrödinger equation.
http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 19
Clicker question 4 Set frequency to AD
x
y
x
y
x
y
Case I: no fixed ends
Case II: one fixed end
Case III: two fixed ends
For which of the three cases do you expect to have only certain frequencies and wavelengths allowed? That is, in which cases will the allowed frequencies be quantized?
A. Case I B. Case II C. Case III D. More than one case
http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 20
Clicker question 4 Set frequency to AD
x
y
x
y
x
y
Case I: no fixed ends
Case II: one fixed end
Case III: two fixed ends
For which of the three cases do you expect to have only certain frequencies and wavelengths allowed? That is, in which cases will the allowed frequencies be quantized?
A. Case I B. Case II C. Case III D. More than one case
After applying the 1st boundary condition we found B=0 but we did not have quantization. After the 2nd boundary condition we found k=nπ/L. This is the quantization.
http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 21
Electron bound in atom Free electron
Only certain energies allowed Quantized energies
Any energy allowed
E
Boundary Conditions ⇒ standing waves
No Boundary Conditions ⇒ traveling waves
Boundary conditions cause the quantization
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Works for light (photons), why doesn’t it work for electrons?
Getting to Schrödinger’s wave equation
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Clicker question 4 Set frequency to AD The equation E = hc/λ is…
A. true for photons and electrons B. true for photons but not electrons C. true for electrons but not photons D. not true for either electrons or photons
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Clicker question 4 Set frequency to AD The equation E = hc/λ is…
A. true for photons and electrons B. true for photons but not electrons C. true for electrons but not photons D. not true for either electrons or photons
works for photons and electrons
works for photons and electrons
only works for massless particles (photons)
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Equal numbers of derivatives result in
doesn’t work for electrons. What does?
Getting to Schrödinger’s wave equation
Note that each derivative of x gives us a k (momentum) while each derivative of t gives us an ω (energy).
or
For massive particles we need
So we need two derivatives of x for p2 but only one derivative of t for K.
If we add in potential energy as well we get the Schrödinger equation…