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### Transcript of Matter waves: Ψmx.nthu.edu.tw/~mingchang/Courses/Lecture_pdf/MP_20.pdf · PDF file...

• Matter waves:

Probability density = P(x,t) = |Ψ|2 = Ψ*Ψ

P(x,t=0)

Ψ(x,t=0)

xL -L

L-L x Probability of electron being in interval dx = P(x)·dx

Wave function = Ψ(x,t)

dx

More general: Probability of finding electron between x1 and x2 at time t: P(x,t)dx = |Ψ(x,t)|

2dx x1

x2

x1

x2

à This requires ‘normalization’ of ψ(x)

• Normalization Probability density = P(x,t) = |Ψ|2 = Ψ*Ψ

P(x,t=0)

L-L x

à “Normalized wave function”

The probability of finding the particle anywhere in space (i.e. between –∞ and +∞) must be 100%!

P(x,t)dx = |Ψ(x,t)|2dx ≡ 1 -∞

-∞

∞ , ∀t

(except if the particle decays!)

• Example for a probability density

Probability density of grades: What is the probability of getting a score between 38-40?

0 20 50

0.06 What is Probability of getting between 42.5 - 45?

Probability = P(x)*dx = 0.07*2.5 = 0.16

P(x)42.5

45

(~16/100 students)

0

2

4

6

8

10

12

14

16

0 2.5 5 7.5 10 12. 5 15 17.

.5 20 22. 5 25 27.

5 30 32. 5 35 37.

5 40 42. 5 45 47.

5 50 Score

Fre que

ncy

• All sort of funky… ØParticles are described by wave functions (Ψ)

(generally Ψ can have both real and imaginary components) Ø|Ψ|2 tells us probability density ØElectron waves ~ waves of probability.

Electron double slit experiment. Display=Magnitude2 of wave function

Large Magnitude |Ψ|2 means: probability of detecting electron here is high

Small Magnitude |Ψ| 2 means: probability of detecting electron here is low

• • The “blob” in Quantum Wave Interference is a 2D wave packet.

• In this simulation, intensity (i.e. |Ψ|2 or |E|2) is represented by brightness.

Electron Photon

|Ψ|2 |Ε|2

• Review: Ψ, a wave of probability

What is the value of normalization constant ‘c’, such that P(x) integrated over all space =1?

A. L B. sqrt(1/L) C. 1/L D. sqrt(2/L) E. 2/L

Example: Ψ(x,t=0) =

0

c

0 L

Ψ(x,t=0)

x

Normalization of ψ: Integral of |ψ|2 over all space must be = 1 (100% likely will find the particle somewhere in space)!

0, for x=0{

• Ψ : a wave of probability Ψ(x,t=0) = 0 for x

• -L

α

0 L

Ψ(x,t=0)

a bdx

Say an electron were described by the following wave function:

How do the probabilities of finding the electron near of a and b compare (within an interval dx)?:

A. a > 2b B. a = 2b C. a < 2b

x

Ψ(x,t=0)= αx/L from -L to L Ψ(x,t=0)=0 elsewhere

P(x,t=0) = |ψ(x,t=0)|2

• 0-L

α

0 L

Ψ(x,t=0)=αx/L from -L to L

a bdx

How do the probabilities of finding the electron near (within dx) of a and b A. a>2b B. a=2b C. a 2 * (P(x) at b) Probability density is what we detect!!

Say an electron were described by the following wave function:

• Vibrations on a string: Electromagnetic waves:

Wave Equations

2

2

22

2 1 t y

vx y

∂ ∂=

∂ ∂

v = speed of wave

2

2

22

2 1 t E

cx E

∂ ∂=

∂ ∂

c = speed of light

x

yE

x

Magnitude is non-spatial: = Strength of Electric field

Magnitude is spatial: = Vertical displacement of String

Solutions: E(x,t) Solutions: y(x,t)

• What are matter waves? EM Waves (light/photons): Amplitude E = electric field E2 tells us probability of

finding photon. Maxwell’s Equations:

Solutions are real sin/cosine waves:

Matter Waves (electrons, etc.): Amplitude ψ = “wave function” |ψ|2 tells us probability of

finding particle. Schrödinger Equation:

Solutions are complex sine/cosine waves:

2

2

22

2 1 t E

cx E

∂ ∂=

∂ ∂

t i

xm ∂ ∂=

∂ ∂− ψψ !! 2 22

2

)cos(),( )sin(),( tkxAtxE tkxAtxE

ω ω −= −=

))sin()(cos( ),( )(

tkxitkxA Aetx tkxi

ωω ψ ω

−+− == −

• Review of sinusoidal waves:

Wave in time: cos(2πt/T) = cos(ωt) = cos(2πf·t)

Wave in space: cos(2πx/λ) = cos(kx)

k is spatial analogue of angular frequency ω. (We use k because it’s easier to write sin(kx) than sin(2πx/λ).)

λ = Wavelength = length of one cycle

k = 2π/λ = wave number = number of radians per meter

x

T = Period = time of one cycle

ω = 2π/T = angular frequency = number of radians per second

t

• Plane Waves • Most general kinds of waves are plane waves

(sines, cosines, complex exponentials) – extend forever in space

• ψ1(x,t) = exp(i(k1x-ω1t))

• ψ2(x,t) = exp(i(k2x-ω2t))

• ψ3(x,t) = exp(i(k3x-ω3t))

• ψ4(x,t) = exp(i(k4x-ω4t))

• etc… Different k’s correspond to different energies, since E = ½mv2 = p2/2m = h2/2mλ2 = !2k2/2m

• Superposition principle • If ψ1(x,t) and ψ2(x,t) are both solutions to

wave equation, so is ψ1(x,t) + ψ2 (x,t). → Superposition principle

• E.g. homework (HW8, Q7b) – superposition of waves one traveling to the left and to the right create a standing wave:

• We can make a “wave packet” by combining plane waves of different energies:

)sin()sin(2)sin()sin(),( tkxAtkxAtkxAtxE ωωω −=+−−= ???

ψ (x,t) = ΣnAnexp(i(knx-ωnt))

à

• 35

Superposition

• Plane Waves vs. Wave Packets Plane Wave: ψ(x,t) = Aexp(i(kx-ωt))

Wave Packet: ψ(x,t) = ΣnAnexp(i(knx-ωnt))

Which one looks more like a particle? • In real life, matter waves are more like wave packets.

Mathematically, much easier to talk about plane waves, and we can always just add up solutions to get wave packet.

• Method of adding up sine waves to get another function (like wave packet) is called “Fourier Analysis.” You will explore it with simulation in the homework.

• 37

• Plane Waves vs. Wave Packets

A. p most well-defined for plane wave, x most well-defined for wave packet.

B. x most well-defined for plane wave, p most well-defined for wave packet.

C. p most well-defined for plane wave, x equally well-defined for both.

D. x most well-defined for wave packet, p most well-defined for both.

E. p and x equally well- defined for both.

Plane Wave: Ψ(x,t) = Aei(kx-ωt) :

Wave Packet: Ψ(x,t) = ΣnAnei(knx-ωnt) :

For which type of wave are position x and momentum p most well-defined?

• Plane Waves vs. Wave Packets Plane Wave: Ψ(x,t) = Aei(kx-ωt)

– Wavelength, momentum, energy: well-defined. – Position: not defined. Amplitude is equal everywhere,

so particle could be anywhere!

Wave Packet: Ψ(x,t) = ΣnAnei(knx-ωnt)

– λ, p, E not well-defined: made up of a bunch of different waves, each with a different λ,p,E

– x much better defined: amplitude only non-zero in small region of space, so particle can only be found there.

• 3

Superposition

• Three deBroglie waves are shown for particles of equal mass.

I II III x

The highest speed and lowest speed are: a. II highest, I & III same and lowest b. I and II same and highest, III is lowest c. all three have same speed d. cannot tell from figures above

ans b. shorter wavelength means larger momentum = larger speed. III is largest wavelength, I and II are same. amplitude of wave is not related to speed.

A, 2f 2A, 2f A, f

xx

Review question

A: Amplitude. f: frequency

• E=hc/λ…

A. …is true for both photons and electrons. B. …is true for photons but not electrons. C. …is true for electrons but not photons. D. …is not true for photons or electrons.

E = hf is always true but f = c/λ only applies to light, so E = hf ≠ hc/λ for electrons.

c = speed of light!

• Today:

• Heisenberg’s uncertainty relation

• Intro to Schrödinger's equation

• Heisenberg Uncertainty Principle • In math: Δx·Δp ≥ !/2 (or better: Δx·Δpx ≥ !/2) • In words: Position and momentum cannot both

be determined precisely. The more precisely one is determined, the less precisely the other is determined.

• Should really be called “Heisenberg Indeterminacy Principle.”

• This is weird if you think about particles. But it’s very clear if you think about waves.

• Heisenberg Uncertainty Principle

Δx small Δp – only one wavelength

Δx large Δp – wave packet made of lots of waves

Δx medium Δp – wave packet made of several waves

• A slightly different scenario:

x

y

Plane-wave propagating in