What are these waves?

Transcription

1 PH300 Modern Phsics SP11 Is the state of the cat to be created onl when a phsicist inestigates the situation at some definite time? Nobod reall doubts that the presence or absence of a dead cat is something independent of obseration. Albert Einstein 4/5 Da 0: Ques/ons? Reiew waes and wae equa/ons Schrodinger equa/on Square well poten/al Thursda: Finite Square Well Quantum Tunneling Models of the Atom Thomson Plum Pudding Wh? Known that negatie charges can be remoed from atom. Problem: just a random guess Rutherford Solar Sstem Wh? Scattering showed hard core. Problem: electrons should spiral into nucleus in ~10-11 sec. Bohr fied energ leels + Wh? Eplains spectral lines. Problem: No reason for fied energ leels + debroglie electron standing waes Wh? Eplains fied energ leels + Problem: still onl works for Hdrogen. Schrödinger will sae the da!! What are these waes? EM Waes (light/photons) Amplitude electric field tells ou the probabilit of detec/ng a photon. Mawell s Equa/ons: Ma)er Waes (electrons/etc) Amplitude maqer field tells ou the probabilit of detec/ng a par/cle. Schrödinger Equa/on: Toda: Work towards finding an equation that describes the probabilit wae for a particle in an situation. Rest of QM: Soling this differential equation for arious cases and using it to understand nature and technolog Look at general aspects of wae equations appl to classical and quantum waes. Solu/ons are sine/cosine waes: Solu/ons are comple sine/cosine waes: A. Start b reiewing some classical wae equations. B. Introduce Schrödinger equation. 4 Schrodinger equation deelopment-- new approach to phsics Old approach- understand phsical sstem well, reason out equations that must describe it. New approach-- see what eperiment shows but not know wh. Write down equation that has sensible basic math properties. See if when soled, gies solutions that match eperiment. Schrödinger starting point-- what do we know about classical waes (radio, iolin string)? Which aspects of a particle wae equation need to be similar and which different from classical wae equations? 5 Electromagnetic waes: E E 1 E c cspeed of light Solutions: E( Magnitude is non-spatial: Strength of electric field Wae Equations Vibrations on a string: speed of wae Solutions: ( Magnitude is spatial: Vertical displacement of string 6 1

2 Wae Equation How to sole? In DiffEq class, learn lots of algorithms for soling DiffEq s. In this class, focus on onl Diff Eq s: ψ ψ k ψ and α ψ (k & α ~ constants) How to sole a differential equation in phsics: 1) Guess functional form for solution ) Make sure functional form satisfies Diff EQ (find an constraints on constants) 1 deriatie: need 1 soln à f(f 1 deriaties: need soln à f( f 1 + f 3) Appl all boundar conditions The hardest part! (find an constraints on constants) 7 1) Guess functional form for solution Which of the following functional forms works as a possible solution to this differential equation? I. ( A t, II. ( Asin(B) III. ( Acos(B)sin(C a. I b. III c. II, III d. I, III e. None or some other combo Does it satisf Diff EQ? Answer is b. Onl: (Acos(B)sin(C General Soln (usuall use k and ω): ( Asin(k)cos(ω + Bcos(k)sin(ω 8 1) Guess functional form for solution II. ( Asin(B) III. ( Acos(B)sin(C ( Asin( B) LHS: AB sin( B) 1 RHS: 0 AB sin( B) 0 sin( B) 0 Not OK! is a ariable. There are man alues of for which this is not true! LHS: ( Acos( B)sin( C AB cos( B)sin( C 1 AC RHS: cos( B)sin( C AC AB cos( B)sin( C cos( B)sin( C C B OK! B and C are constants. 9 Constrain them so satisf this. υ speed of wae What functional form works? Two eamples: (Asin(k)cos(ω + Bcos(k)sin(ω (Csin(k-ω + Dsin(k+ω k, ω, A, B, C, D are constants ( C sin( k ω Cω Ck sin( k ω sin( k ω ω ω Satisfies wae eqn if: k ν fλ k 10 (Asin(k)cos(ω + Bcos(k)sin(ω (Csin(k-ω + Dsin(k+ω t0 What is the waelength of this wae? Ask ourself à How much does need to increase to increase k-ωt b π? sin(k(+λ) ω sin(k ωt + π) k(+λ)k+π kλπ è kπ/ λ kwae number (radians-m -1 ) Boundar conditions? l. ( 0 at 0 and L 0 L Functional form of solution? ( Asin(k)cos(ω + Bcos(k)sin(ω At 0: ( Bsin(ω 0 ( Asin(k)cos(ω à onl works if B0 What is the period of this wae? Ask ourself à How much does t need to increase to increase k-ωt b π? sin(k-ω (t+t)) sin(k ωt + π ) Speed ωtπ è ωπ/t ω angular frequenc λ ω πf T k 11 Ealuate (0 at L. What are possible alues for k? a. k can hae an positie or negatie alue b. π/(l), π/l, 3π/(L), π/l Answer is d: knπ/l c. π/l Boundar conditions put d. π/l, π/l, 3π/L, 4π/L constraints on k 1 e. L, L/, L/3, L/4,. causes quantization of k and λ!!!

3 0 L With Wae on Violin String: Find: Onl certain alues of k (and thus λ) allowed à because of boundar conditions for solution Which boundar conditions need to be satisfied? I. ( 0 at 0 and L ( Asin(k)cos(ω +Bcos(k)sin(ω n1 Same as for electromagnetic wae in microwae oen: At 0: Bsin(ω 0 à B0 At L: Asin(kL)cos(ω 0 à sin(kl)0 à kl nπ (n1,,3, ) à knπ/l ( Asin(nπ/L)cos(ω n n3 13 Eactl same for electrons in atoms: Find: Quantization of electron energies (waelengths) à from boundar conditions for solutions to Schrodinger s Equation. 14 Three strings: Case I: no fied ends Three strings: Case I: no fied ends Case II: one fied end Case II: one fied end Case III, two fied end: Case III, two fied end: For which of these cases, do ou epect to hae onl certain frequencies or waelengths allowed that is for which cases will the allowed frequencies be quantized. For which of these cases, do ou epect to hae onl certain frequencies or waelengths allowed that is for which cases will the allowed frequencies be quantized. a. I onl b. II onl c. III onl d. more than one a. I onl b. II onl c. III onl d. more than one 15 Quantization came in when applied nd boundar condition, 16 bound on both sides Electron bound in atom (b potential energ) PE Free electron Schrodinger s starting point: What do we know about classical waes (radio, iolin string)? What aspects of electron wae eq n need to be similar and what different from those wae eqs? Onl certain energies allowed Quantized energies An energ allowed Not going to derie it, because there is no deriation Schrodinger just wrote it down. Instead, gie plausibilit argument. Boundar Conditions è standing waes No Boundar Conditions è traeling waes

4 E 1 E c Works for light, wh not for an electron? simple answer-- not magic but details not useful to ou. Adanced formulation of classical mechanics. Each p, 1 partial deriatie with respect to. Each E, 1 partial deriatie with respect to time. light: E (pc), so equal number deriaties and t. (KE) (PE) electron: E p /m +V, so need 1 time deriatie, deriaties with respect to plus term for potential energ V. so finall... + V ( i m Of course for an real sstem, need in 3 dimensions, è just add partial deriaties of and z, and V(,z) etc. Schrödinger wrote it down, soled for hdrogen, got solutions that gae eactl the same electron energ leels as Bohr V ( i m Can differ from book to book: Man gie an equation with no t s in it!! Some use U() we use V() Aboe is correct equation for describing electron as produced b Schrödinger. We will work out a simplified ersion that leaes out the time dependence. 1 Ver important question for a modern engineer: Nanotechnolog: how small does an object hae to be before behaior of electrons starts to depend on size and shape of the object due to quantum effects? How to start? Need to look at: A. Size of wire compared to size of atom. B. Size of wire compared to size of electron wae function. C. Energ leel spacing compared to thermal energ, kt. D. Energ leel spacing compared to some other quantit. (What?) E. Something else. (What?) Nanotechnolog: how small does a wire hae to be before moement of electrons starts to depend on size and shape due to quantum effects? How to start? Need to look at C. Energ leel spacing compared to thermal energ, kt. Almost alwas focus on energies in QM. Electrons, atoms, etc. hopping around with random energ kt. kt >> than spacing, spacing irreleant. Smaller, spacing big deal. So need to calculate energ leels. pit depth compared to kt? 3 + V ( i m Want to use this to calculate electron waes in a phsical situation. First step is: A. Figure out how man electrons will be interacting B. Figure out what general solutions will be b plugging in trial solutions and seeing if can sole. C. Figure out what the forces will be on the electron in that phsical situation. D. Figure out what the boundar conditions must be on the electron wae. E. Figure out what potential energ is at different and t for the phsical situation. 4 4

5 + V ( i m Want to use to calculate electron waes in a phsical situation. First step is: E. Figure out what potential energ is at different and t for the phsical situation. Alread know m is mass of electron, so all need to know is the potential energ V(, which completel determines the situation, and how electron will behae! Need V( for small wire to answer our basic question. Warm up with just a proton. 5 + V ( i m what is V(r, for electron interacting + - with proton? A. -ke /r, where r is the distance from electron to origin. B. -ke /r where r is distance between + and -. C. Impossible to tell unless we know how electron is moing, because that determines the time dependence. D. (-ke /r) sin(ω E. Something else Ans: B - Although potential energ will be different as electron moes to different distance, at an gien distance will be same for all time. So V(r, V(r) -ke /r. H atom. 6 + V ( i m Most phsical situations, like H atom, no time dependence in V! Simplification #1:V V() onl. (works in 1D or 3D) (Important, will use in all Shrödinger equation problems!!) separates into position part dependent part ψ() and time dependent part ϕ( ep(-iet/ħ). ψ()ϕ( Plug in, get equation for ψ() Seems like a good eercise for HW. ψ ( ) + V ( ) ψ ( ) Eψ ( ) m time independent Schrodinger equation 7 ψ ( ) + V ( ) ψ ( ) Eψ ( ) m 1. Figure out what V() is, for situation gien.. Guess or look up functional form of solution. 3. Plug in to check if ψ s and all s drop out, leaing an equation inoling onl a bunch of constants. 4. Figure out what boundar conditions must be to make sense phsicall. 5. Figure out alues of constants to meet boundar conditions and normalization: ψ() d 1-6. Multipl b time dependence ϕ( ep(-iet/ħ) to hae full 8 solution if needed. STILL TIME DEPENDENCE! Where does the electron want to be? The place where V() is lowest. V() Electrons alwas tend toward the position of lowest potential energ, just like a ball rolling downhill. 0 L ψ ( ) + V ( ) ψ ( ) Eψ ( ) m Before tackling wire, understand simplest case. Soling Schrod. equ. Electron in free space, no electric fields or grait around. 1. Where does it want to be? 1. No preference- all the same.. What is V()?. Constant. 3. What are boundar conditions on ψ()? 3. None, could be anwhere. Smart choice of constant, V() 0! ψ ( ) Eψ ( ) m 30 5

6 ψ ( ) Eψ ( ) m What does this equation describe? A. Nothing phsical, just a math eercise. B. Onl an electron in free space along the -ais with no electric fields around. C. An electron fling along the -ais between two metal plates with a oltage between them as in photoelectric effect. D. An electron in an enormousl long wire not hooked to an oltages. E. More than one of the aboe. Ans: E Both (B) and (D) are correct. No electric field or oltage means potential energ constant in space and time, V0. 31 A solution to this differential equation is: (A) A cos(k) (B) A e -k (C) A sin (k) (D) (B & C) (E) (A & C) ψ ( ) Eψ ( ) m Ans: E Both (A) and (C) are solutions. m k Acosk EAcosk solution if k m E 3 ψ ( ) Eψ ( ) m k m E makes sense, because p k Condition on k is just saing that (p )/m E. V0, so E KE ½ m p /m ψ ( ) Acos k The total energ of the electron is: A. Quantized according to E n (constan n, n 1,, 3, B. Quantized according to E n const. (n) C. Quantized according to E n const. (1/n ) D. Quantized according to some other condition but don t know what it is. E. Not quantized, energ can take on an alue. ψ ( ) Acos k k m E p k k (and therefore E) can take on an alue. Almost hae a solution, but remember we still hae to include time dependence: ψ ( ) φ( φ( e iet / bit of algebra, using identit: e i cos() + i sin() Acos(k ω + i Asin(k ω Ans: E - No boundar, energ can take on an alue electron in free space or long wire with no oltage Acos(k ω + Aisin(k ω if k positie 0 L Using equation, probabilit of electron being in d at L is probabilit of being in d at 0. A. alwas bigger than B. alwas same as C. alwas smaller than D. oscillates up and down in time between bigger and smaller E. Without being gien k, can t figure out Ans: B - Prob ~ ψ*ψ A cos (k-ω +A sin (k-ω A, so constant and equal, all t. Which free electron has more kinetic energ? A) 1 B) C) Same big k big KE 1.. small k small KE if V0, then E Kinetic Energ. So first term in Schöd. Eq. is alwas just kinetic energ! ψ() Eψ() m Curature è KE 6