Professor Jasper Halekas Van Allen 70 MWF 12:30-1:20 Lecture

Transcription

1 Professor Jasper Halekas Van Allen 70 MWF 1:30-1:0 Lecture

2 Back on regular schedule for the next two weeks There *will* be labs and homeworks due this week and next

3 EM Waves (light/photons) Amplitude E = electric field E tells you the probability of detec;ng a photon. Maxwell s Equa;ons: E 1 E = x c t Solu;ons are sine/cosine waves: Ma)er Waves (electrons/etc) Amplitude Ψ = mafer field Ψ tells you the probability of detec;ng a par;cle. Schrödinger Equa;on:! Ψ Ψ +U(x,t)Ψ = i! m x t Solu;ons are complex sine/ cosine waves:

4 EM Waves Wave Functions Ext (, ) = Asin( kx ωt) Ext (, ) = Acos( kx ωt) ( ) [ cos( ω ) sin( ω )] Ψ (,) xt = Aexpi kx ωt = A kx t + i kx t In either case, wave packets can be formed by superposing waves with different k values.

5 E y Electromagnetic waves: E = x 1 c E t c=speed of light x Vibrations on a string: y = x 1 v y t v=speed of wave x Solutions: E(x,t) Magnitude is non-spatial: = Strength of electric field Solutions: y(x,t) Magnitude is spatial: = Vertical displacement of string

6 How to solve a differential equation in physics: 1) Guess functional form for solution ) Make sure functional form satisfies Diff EQ (find any constraints on constants) 1 derivative: need 1 soln à f(x,t)=f 1 derivatives: need soln à f(x,t) = f 1 + f 3) Apply all boundary conditions (find any constraints on constants) The hardest part!

7 y x = 1 v y t 1) Guess functional form for solution Which of the following functional forms works as a possible solution to this differential equation? I. y(x, t) = Ax t, II. y(x, t) = Asin(Bx) III. y(x,t) = Acos(Bx)sin(Ct) a. I b. III c. II, III d. I, III e. None or some other combo

8 y x = 1 v y t 1) Guess functional form for solution Which of the following functional forms works as a possible solution to this differential equation? I. y(x, t) = Ax t, II. y(x, t) = Asin(Bx) III. y(x,t) = Acos(Bx)sin(Ct) a. I b. III c. II, III d. I, III e. None or some other combo

9 II. y(x, t) = Asin(Bx) y( x, t) = Asin( Bx) LHS: RHS: y = AB x 1 y v t 1) Guess functional form for solution AB sin( Bx) = 0 sin( Bx) = 0 = III. y(x,t) = Acos(Bx)sin(Ct) y( x, t) = Acos( Bx)sin( Ct) sin( Bx) y LHS: x 0 AB = AB cos( Bx)sin( Ct) 1 y AC RHS: = cos( Bx)sin( Ct) v t v AC cos( Bx)sin( Ct) = v C B = v cos( Bx)sin( Ct)

10 y(x,t)=asin(kx)cos(ωt) + Bcos(kx)sin(ωt) y(x,t)=csin(kx-ωt) + Dsin(kx+ωt) v k ω = = = ) sin( ) sin( ), ( t kx Ck t kx C t x y ω ω Satisfies wave eqn if: ) sin( t kx v C ω ω λ ω ν f k = = 1 t y v x y =

11 y(x,t)=csin(kx-ωt) + Dsin(kx+ωt) What is the wavelength of this wave? Ask yourself àhow much does x need to increase to increase kx-ωt by π? sin(k(x+λ) ωt) = sin(kx ωt + π) k(x+λ)=kx+π kλ=π è k=π/ λ t=0 y k=wave number (radians-m -1 ) What is the period of this wave? Ask yourself àhow much does t need to increase to increase kx-ωt by π? sin(kx-ω (t+τ)) = sin(kx ωt + π ) Speed ωτ=π è ω=π/τ λ ω v = = = πf ω= angular frequency T k x

12 y(x,t) = Asin(kx)cos(ωt) + Bcos(kx)sin(ωt) Boundary conditions? l. y(x,t) = 0 at x=0 and x=l At x=0: y(x,t) = Bsin(ωt) = 0 y(x,t) = Asin(kx)cos(ωt) à only works if B=0 What are possible values for k? a. k can have any positive or negative value b. π/(l), π/l, 3π/(L), π/l c. π/l d. π/l, π/l, 3π/L, 4π/L e. L, L/, L/3, L/4,. 0 L

13 y(x,t) = Asin(kx)cos(ωt) + Bcos(kx)sin(ωt) Boundary conditions? l. y(x,t) = 0 at x=0 and x=l At x=0: y(x,t) = Bsin(ωt) = 0 y(x,t) = Asin(kx)cos(ωt) à only works if B=0 What are possible values for k? a. k can have any positive or negative value b. π/(l), π/l, 3π/(L), π/l c. π/l d. π/l, π/l, 3π/L, 4π/L e. L, L/, L/3, L/4,. 0 L

14 Which boundary conditions need to be satisfied? I. y(x,t) = 0 at x=0 and x=l y(x,t) = Asin(kx)cos(ωt) +Bcos(kx)sin(ωt) At x=0: y = Bsin(ωt) = 0 à B=0 At x=l: y= Asin(kL)cos(ωt)= 0 à sin(kl)=0 à kl = nπ (n=1,,3, ) à k=nπ/l y(x,t) = Asin(nπx/L)cos(ωt) 0 L n=1 n= n=3

15 With Wave on Violin String: Find: Only certain values of k (and thus λ) allowed à because of boundary conditions for solution Same as for electromagnetic wave in microwave oven: Exactly same for electrons in atoms:

16 Three strings: Case I: no fixed ends y x Case II: one fixed end Case III, two fixed end: x x For which of these cases, do you expect to have only certain frequencies or wavelengths allowed that is for which cases will the allowed frequencies be quantized. a. I only b. II only c. III only d. more than one

17 Three strings: Case I: no fixed ends y x Case II: one fixed end Case III, two fixed end: x x For which of these cases, do you expect to have only certain frequencies or wavelengths allowed that is for which cases will the allowed frequencies be quantized. a. I only b. II only c. III only d. more than one

18 Electron bound in atom (by potential energy) PE Free electron Only certain energies allowed Quantized energies Boundary Conditions è standing waves Any energy allowed No Boundary Conditions è traveling waves

19 A confined electron m in a box has wave numbers k = nπ/l. What are the allowed momenta and energy of the particle? A. p = nh/(l), E = nhc/(l) B. p = nh/(l), E = n h /(8m e L ) C. p = hl/(nπ), E = hcl/(nπ) D. p = hl/(nπ), E = h L /(n π )

20 A confined electron m in a box has wave numbers k = nπ/l. What are the allowed momenta and energy of the particle? A. p = nh/(l), E = nhc/(l) B. p = nh/(l), E = n h /(8m e L ) C. p = hl/(nπ), E = hcl/(nπ) D. p = hl/(nπ), E = h L /(n π )

21 At a finite boundary, a wave has to be continuous in both value and slope At an infinite boundary, a wave can be discontinuous in slope but still must be continuous in value

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24 Incident, Reflected, Total Transmitted