Lecture 10 - Chapter. 4 Wave & Particles II
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1 Announcement Course webpage ttp://igenergy.pys.ttu.edu/~slee/40/ Textbook PHYS-40 Lecture 0 HW3 will be announced on Tuesday Feb. 9, 05 Outline: Lecture 0 - Capter. 4 Wave & Particles II Matter beaving as Waves A Double-Slit Experiment (watc video ) Properties of Matter Waves Te Free-Particle Scrödinger Equation Uncertainty Principle Te Bor Model of te Atom Matematical Basis of te Uncertainty Principle Te Fourier Transform 4. Properties of Matter Waves (Q) Wat properties caracterize a wave? (A) λ, f, v, also A tat varies wit position/time (i.e. (x,) Wave function (WF): we use different symbol for WF of different kinds of wave () For transverse wave on string; y(x, -- string s transverse displacement () For EM plane wave; E(x, &B(x, -- describe ow te oscillating E- & B-field vary wit (x, (3) For matter wave; ψ(x, = probability amplitude ψ(x,! tell us probability of finding te particle (see next slide for some details)
2 4. Properties of Matter Waves Wave lengt π λ = = De Broglie s ypotesis: "=! λ C = k p mc λ = Wavelengt of matter wave - Experimentally confirmed; e.g. x-tal diffraction de Broglie got Nobel prize in 99 Frequency (of matter waves) f = E/ now, it s open more convenient to express λ = /p and f = E/ in terms of te following quantities; k = π/λ (wave number), w = π/t (angular frequency) Anoter convenient definition: = /π =.055 x 0-34 Js express te fundamental wave-particle relationsip as! p = " k! E =!ω Wavelengt Frequency Properties of Matter Waves De Broglie (99)! = f = p E Amplitude = function of (x, Properties of Matter Waves! Probability Amplitude Amplitude = function of (x, Te probability density to find a particle at coordinate x, at time t! Probability Amplitude!
3 Te probability to find a particle in an interval "x, "t x P dxdt! =!! Integrate over "x, "t t P ( x) dt! =! Properties of Matter Waves Speed v = f! BUT v is NOT te velocity of te massive particle, it s te velocity of te Matter wave! = p
4 ! = p Wy! is so Small? Te probability Last to find Lecture: a particle in an interval "x, "t P dxdt! =!! Integrate over "x, "t Because is very-very Small How does te Probability Wave Move? Equation of Motion for #? Te Free-Particle Scrodinger Wave Equation Analogy: [Q] How do we determine te ψ(x, of a matter wave? Waves on a string wave speed Waves on a string wave speed Solution = function Plane Wave
![4.3 Te Free-Particle Scrodinger Eq. [Q] How do we determine te ψ(x, of a matter wave? [A] Matter Waves -- wave eq. obeyed by matter waves!scrödinger eq.](/docs-images/85/92273902/images/5-0.jpg "-- for free particle, in te absence of external forces, Te Free-Particle Scrodinger Wave Equation [Q] Scrodinger eq. is complex. Matter wave is not real?")
![[A] Reason for i :: NOT matter waves are unreal BUT tey can t be represented by a single real function!! real now!! e.g. like EM waves, parts (E & B) and a complex function, carrying twice te info.](/docs-images/85/92273902/images/5-1.jpg "of a real one.")
![***** Probability Density --- Probability of detecting te particle ~ (wave s Amplitute) [Q] Wat does tis mean if wave as parts; E & B or real & imaginary part of ψ(x,? 4.](/docs-images/85/92273902/images/5-3.jpg "3 Te Free-Particle Scrödinger Eq. Te Plane Wave [Solution] -- complex exponential! x, \"! + i! ) ( t *! x, \"! # i! ( Complex Conjugate i \" i # = # Probability Density =?")
5 4.3 Te Free-Particle Scrodinger Eq. [Q] How do we determine te ψ(x, of a matter wave? [A] Matter Waves -- wave eq. obeyed by matter waves!scrödinger eq. -- for free particle, in te absence of external forces, Te Free-Particle Scrodinger Wave Equation [Q] Scrodinger eq. is complex. Matter wave is not real? [A] Reason for i :: NOT matter waves are unreal BUT tey can t be represented by a single real function!! real now!! e.g. like EM waves, parts (E & B) and a complex function, carrying twice te info. of a real one. ===> In EM, we could treat E & B as a single complex unit by including i w/o making eiter field unreal ===> Combined WF, G = E + icb (Wave Function) = Probability Density! *!,! = ( x!! Probability Wave Function Probability Density a complex function 4.3 Te Free-Particle Scrödinger Eq. ***** Probability Density --- Probability of detecting te particle ~ (wave s Amplitute) [Q] Wat does tis mean if wave as parts; E & B or real & imaginary part of ψ(x,? 4.3 Te Free-Particle Scrödinger Eq. Te Plane Wave [Solution] -- complex exponential! x, "! + i! ) ( t *! x, "! # i! ( Complex Conjugate i " i # = # Probability Density =? probability of finding te particle in certain region!!! x, "! + i! ) ( t *! x, "! # i! ( Probability density = ψ(x, Probability Density *! ( x, =!! = i = =!! # ii!! # i!! + i!! = =!! +!!! ( x, =! +! # per unit lengt per unit volume
6 Te Plane Wave Is te Plane Wave Ue! Is te Plane Wave Is te Plane Wave Taking te partial derivatives on bot sides..
7 Is te Plane Wave Is te Plane Wave Let s see ow te Scrödinger wave eq. relates to te classical pysics of particle? YES So. Our answer is YES. ( mv) E It s = a solution of = m Scrödinger eq. mv Is te Plane Wave Is te Plane Wave YES Mean: Particle s KE = Total E True, classically!! since a free particle as no PE ( mv) E = = m mv Te Scrödinger equation is related to a classical accounting of energy!! YES Mean: Particle s KE = Total E True, classically!! since a free particle as no PE ( mv) E = = m mv
8 Te Magnitude of a Plane Wave Te Magnitude of a Plane Wave Constant in space and time! Constant in space and time! Constant Probability Density Kinetic energy Plane wave = constant (x,
9 Uncertainty Principle
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