Getting to the Schrödinger equationjcumalat/phys2170_f13/lectures/Lec22.pdf · Getting to Schrödinger’s wave equation Note that each derivative of x gives us a k (momentum) while

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)

http://www.colorado.edu/physics/phys2170/

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

http://www.colorado.edu/physics/phys2170/

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


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:


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?


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.


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)


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:


A.  B.  C.  D.  E.  More than one of the above

Clicker question 2 Set frequency to AD


Step 1: Guessing the functional form of the solution

Which of the following function forms is a possible solution to this differential equation?


A.  B.  C.  D.  E.  More than one of the above



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.


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


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.


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


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, ….


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:


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


Standing waves

Standing wave

Standing wave constructed from two traveling waves moving in opposite directions


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.



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



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.


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


Works for light (photons), why doesn’t it work for electrons?

Getting to Schrödinger’s wave equation


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


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)


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…

Documents

Getting to the Schrödinger equationjcumalat/phys2170_f13/lectures/Lec22.pdf · Getting to Schrödinger’s wave equation Note that each derivative of x gives us a k (momentum) while