PHYSICS 4E FINAL EXAM SPRING QUARTER 1 Fomulas and constants: hc =1,4 ev A ; k B =1/11,6 ev/k ; ke =14.4eVA ; m e c =.511"1 6 ev ; m p /m e =1836 Relativistic enegy - momentum elation E = m c 4 + p c ; c = 3!1 8 m/ s Photons: E = hf ; p = E /c ; f = c /" Loentz foce: F = qe + qv " B Photoelectic effect : ev = ( 1 mv ) max = hf " #, # wok function di Integals : I n " & x n e #x dx ; n = #I n + ; I = 1 ' ; I 1 = 1 d ; x & 3 dx = ' 4 e x #1 15 Planck's law : u(") = n(") E _ (") ; n(") = 8# " ; 4 E_ (") = hc 1 " e hc / "kbt 1 Enegy in a mode/oscillato : E f = nhf ; pobability P(E) "e #E / k B T Stefan's law : R = "T 4 ; " = 5.67 #1 8 W /m K 4 ; R = cu /4, U = u()d Wien's displacement law : " m T = hc /4.96k B Compton scatteing : "'-" = h m e c (1# cos) ; " c h m e c =.43A Ruthefod scatteing: b = kq " Q m " v cot(# /) ; "N # 1 sin 4 ( /) Electostatics : F = kq q 1 (foce) ; U = q V (potential enegy) ; V = kq (potential) Hydogen spectum : 1 " = R( 1 m # 1 n ) ; R =1.97 1 17 m #1 = 911.3A Boh atom: n = n ; = a Z ; E Z n = "E n ; a = h mke =.59A ; E = ke =13.6eV ; L = mv = nh a E k = 1 mv ; E p = " ke Z Reduced mass : µ = ; E = E k + E p ; F = ke Z = m v & ' ; hf = hc/# = E n " E m mm m + M ; X - ay specta : f 1/ = A n (Z " b) ; K : b =1, L : b = 7.4 de Boglie : " = h p ; f = E h ; # = f ; k = p ; E = h# ; p = hk ; E = ; hc =1973 ev A " m goup and phase velocity : v g = d" ; v p = " ; Heisenbeg : #x#p ~ h ; #t#e ~ h dk k Wave function "(x,t) = "(x,t) e i# (x,t ) ; P(x,t) dx = "(x,t) dx = pobability Schodinge equation : - h m " # " # +V(x)#(x,t) = ih "x " t Time " independent Schodinge equation : - h m # #x " squae well: # n (x) = L sin(nx L ) ; E n = h n ; x ml op = x, p op = h i x ; #(x,t) =(x)e -i E h t +V(x)(x) = E(x) ; dx & * =1 - " ; < A >= & dx# * A op # Eigenvalues and eigenfunctions: A op " = a " (a is a constant) ; uncetainty : #A = < A > < A > Hamonic oscillato : " n (x) = C n H n (x)e # m h x ; E n = (n + 1 p )h ; E = m + 1 m x = 1 m A ; n = ±1 -"
PHYSICS 4E FINAL EXAM SPRING QUARTER 1 Step potential: R = (k " k 1 ) m, T =1" R ; k = (E "V) (k 1 + k ) h Tunneling : "(x) ~ e -#x ; T ~ e -#x ; T ~ e b - #(x )dx a ; #(x) = m[v (x) - E] h 3D squae well: "(x,y,z) = " 1 (x)" (y)" 3 (z) ; E = # h m (n 1 L + n 1 L + n 3 L ) 3 Spheically symmetic potential: " n,l,m (,#,) = R nl ()Y lm (#,) ; Y lm (#,) = f lm (#)e im Angula momentum : L = " p ; L z = h # i # ; L Y lm = l(l +1)h Y lm ; L z = mh Radial pobability density : P() = R n,l () ; Enegy : E n =!13.6eV Z Gound state of hydogen and hydogen - like ions : " 1,, = 1 # 1/ ( Z a ) 3 / e Z / a Obital magnetic moment : µ " = #e m e L " ; µ z = #µ B m l ; µ B = eh m e = 5.79 1 #5 ev /T Spin 1/ : s = 1, S = s(s +1)h ; S = m h ; m = ±1/ ; z s s µ s = "e gs m e Total angula momentum: J = L + S ; J = j( j +1)h ; l " s # j # l + s ; " j # m j # j Obital +spin mag moment : µ = "e m ( L + gs ) ; Enegy in mag. field : U = " µ # B Two paticles : "(x 1,x ) = + /# "(x, x 1 ) ; symmetic/antisymmetic Sceening in multielecton atoms : Z! Z eff, 1 < Z eff < Z Obital odeing: 1s < s < p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 6d ~ 5f Boltzmann constant : k B =1/11,6 ev/k 1 f B (E) = Ce "E / kt ; f BE (E) = e # e E / kt "1 ; f (E) = 1 FD e # e E / kt +1 ; n(e) = g(e) f (E) Rotation : E R = L I, I = µr, vibation : E v = h"(# + 1 ), " = k /µ, µ = m 1 m /(m 1 + m ) g(e) = ["(m) 3/ V /h 3 ]E 1/ (tanslation, pe spin) ; Equipatition : < E >= k B T / pe degee of feedom Justify all you answes to all poblems. Wite clealy. n
PHYSICS 4E FINAL EXAM SPRING QUARTER 1 Poblem 1 (1 pts) electon electon Conside an electon in an infinite squae well potential and an electon in a hamonic oscillato potential, both in 1 dimension. Both electons ae in the lowest enegy state. The longest wavelength photons that these electons can absob has wavelength 1A. (a) Find the width of the squae well, in A. (b) Find the classical amplitude of oscillation fo the electon in the gound state of this hamonic oscillato potential, in A. (c) Compae the pobabilities that the electon is at the cente of the well in both cases. Which one is lage? (d) Daw the wavefunctions fo both cases and point out thei similaities and diffeences. & Hints: e "#x dx = # ; h /m e = 7.6 ev A " Poblem (1 pts) A paticle moving in one dimension is descibed by the wavefunction "(x) = Cxe #x /(x ) with C and x constants. (a) Calculate the uncetainty in the position of this paticle, Δx. Give you answe in tems of x. (b) At what value of x is this paticle most likely to be found? Give you answe in tems of x. (c) Appoximately how much moe o less likely is it to find this paticle at position.x than it is to find it at position.1x? 1 & Hint: x e "#x dx = # ; x 4 e "#x 3 & dx = 3 4 # 5 " " Poblem 3 (1 pts) An electon moves in the one-dimensional potential # 1" cos(x /a)& V (x) = V ( 1+ cos(x /a)' fo "#a < x < #a, V (x) = " othewise. a=1a. The electon is descibed by the wavefunction: "(x) = C(1+ cos(x /a)) fo "#a < x < #a, "(x) = othewise. C is a constant. (a) Find the values of V and of the enegy E so that this wavefunction is a solution of the Schodinge equation fo this potential. Give you answes in ev. (b) Find the classically allowed egion fo this electon. I.e., the egion whee the potential enegy of the electon is smalle than its total enegy. (c) Make a gaph of the potential vesus x and of the wavefunction vesus x, showing in paticula whee the classical tuning points ae, whee the second deivative of the wavefunction is negative and whee it is positive, whee the wavefunction goes to zeo and whee the potential goes to ".
PHYSICS 4E FINAL EXAM SPRING QUARTER 1 Poblem 4 (1 pts) Conside a solid as a system of N thee-dimensional hamonic oscillatos, each of them has fequency " =.ev/h. As you know, its heat capacity C V at high tempeatues is 3Nk B, with k B =Boltzmann's constant (Doulong-Petit law). Find the enegy of this system at tempeatues (a) 1K (b) 1K (c) 1K Give you answes as CNk B T, with C a numeical coefficient. Poblem 5 (1 pts) An electon in a hydogen-like ion is descibed by the wavefunction "(,#,) = Ce / a cos& with C a constant and a the Boh adius. (a) Give the values of the quantum numbes n, l, m and of the ionic chage Z. Justify you answes. (b) Find the most pobable value fo this electon. (c) Calculate the aveage value of 1/ fo this electon. (d) Compae you esults in (b) and (c) with the pedictions of the Boh atom. Use d s e "# = s! #. s+1 Hint: you need to nomalize the pobability distibution. Poblem 6 (1 pts) e 9 ev 1A 6A An electon is in the gound state of the one-dimensional potential well shown. Estimate how long it will take it to escape, in seconds. Hint: compute the tunneling pobability and the numbe of attempts pe second using the velocity and width of the well.
PHYSICS 4E FINAL EXAM SPRING QUARTER 1 Poblem 7 (1 pts) Conside a gas of hydogen atoms whee all the electons ae initially in the n=3, l =1 state. (a) What ae the wavelengths of the photons emitted when these electons make tansitions to lowe enegy states? Ignoe spin-obit coupling. (b) Assume all the electons ae in the n=3, l =1 state, and a magnetic field of magnitude 1T is tuned on. Ignoe spin-obit coupling. What ae the wavelengths of the photons emitted when these electons make a tansition to the gound state? (c) Assume all the electons ae in the n=3, l =1 state, no magnetic field, but take into account spin-obit coupling. Assume the effective magnetic field seen by the electon is B eff =.5T. Give the possible values of the quantum numbe J, the change in enegy fo each J due to spin-obit coupling, and indicate which J gives the highe enegy and why. Poblem 8 (1 pts) Conside a two-dimensional hamonic oscillato potential V (x, y) = 1 K(x + y ) The Schodinge equation fo a paticle of mass m moving in two dimensions in this potential is sepaable into one-dimensional equations, just like the Schodinge equation fo a two-dimensional infinite well is sepaable into one-dimensional equations. (a) Wok out the math to show that the above statement is tue, and explain. (b) Give the expession fo the gound state wavefunction of a paticle in this potential, "(x, y), in tems of m and " = K /m. (c) Give the expession fo the enegy of the fist excited state of this paticle. What is its degeneacy? What ae the possible quantum numbes? (d) If thee ae 4 identical femions of mass m and the same spin in this two-dimensional hamonic oscillato potential, what is the total enegy of the system? Give you answe in tems of ω. Justify all you answes to all poblems. Wite clealy.