Quantum Physics Lecture 6
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1 Quantum Physics Lecture 6 Bohr model of hydrogen atom (cont.) Line spectra formula Correspondence principle Quantum Mechanics formalism General properties of waves Expectation values Free particle wavefunction 1-D Schroedinger Equation
2 Experimental Evidence for Bohr model From optical emission spectrum of hydrogen: Consists of line spectra (in contrast to blackbody continuum) Balmer series; fitted to formula 1 λ = R( ) n 2 for n = 3, 4, 5... Experimental value of R = x 10 7 m -1 In general, use two integers n i (initial) and n f (final) 1 λ = R 1 2 n 1 2 f n i for n i > n f Where n f = 1 (Lyman), = 2 (Balmer), = 3 (Paschen) etc
3 Connect expt. with Bohr model 1 λ = R 1 2 n 1 2 f n i E n = me 4 ( ) 8ε o 2 h 2 1 n 2 Optical emission: result of a down transition of electron from a higher energy orbit (n i ) to a lower energy orbit (n f ) Energy difference is emitted as a photon: ω = E ni E n f = me4 1 8ε 2 0 h 2 2 n 1 2 i n f = hc λ 1 λ = me4 8ε 0 2 h 3 c 1 n f 2 1 n i 2 R = me4 8ε 2 0 h 3 c
4 Centre-of-mass correction R = me4 8ε o 2 h 3 c =1.097x107 m 1 This value of R agreed with expt. of the time! However, later (more accurate) experiments gave R expt = whereas model gave R = Replace m with m* = mm/(m + M) = (m) In centre-of-mass description of electron (m) and proton (M) Correspondence Principle The greater the quantum number. the closer Quantum Physics approaches Classical Physics!
5 Correspondence Principle Compare orbit frequency (f) and emitted photon frequency (ω/2π) f = v 2πr = e 2π 4πε o mr = me ε 2 o h 3 ν = ω/2π = E/h me4 1 8ε 2 o h 3 2 n 1 2 f n i Note same pre-factor! Write n i = n and n f = n - p n 3 1 n f 2 1 n i 2 = 1 ( n p) 2 1 n 2 = When n >> p, (2np - p 2 ) ~ 2np and (n - p) 2 ~ n 2 then ω/2π ~pf and letting p=1, a transition: n to (n-1) gives ω/2π ~ f 2np p 2 ( n p) 2 n 2 2 p n 3
6 Bohr criterion for allowed orbits The Bohr requirement for orbits can also be stated as: angular momentum is quantised in units of ħ mvr = n h 2π ( ) = n 2πr = n h mv ( ) = nλ i.e. Equivalent to fitting de Broglie wavelengths In fact, the concept of quantised angular momentum nħ is the fuller and broader criterion, with further consequences to be seen; the other is merely illustrative, not factual! Complete description of atom requires at least three different quantum numbers (see later lectures)
7 General properties of waves Recall 1-D wave: y = Acos(ωt - kx) This is just one possible solution of the 1-D wave equation: δ 2 y δx = 1 δ 2 y 2 v 2 δt 2 [ ( )] where i = 1 ( ) isin( ωt kx) - the general (complex) solution. In general y = Aexp i ωt kx [ ] = A cos ωt kx For waves (of existence) in QM use wavefunction ψ Recall UP and probability: y 2 is a measure of probability of finding a particle at location x In QM, ψ is in general complex, and not usually a measureable parameter (such a momentum etc.) However, ψ 2 is! So must retain full complex solution, not just real part.
8 General properties of waves cont. Three other required properties of wavefunction ψ (1) single-valued and continuous (2) derivative (dψ/dx) single-valued and continuous (3) normalisable: ψ 2 dx = 1 - i.e. integrated probability density over all space is unity i.e. for 1-D case require If ψ is complex, what about ψ 2? ψ 2 = ψ*ψ where ψ* is complex conjugate of ψ ψ = A + ib and ψ* = A - ib ψ 2 dx = 1 ψ*ψ = (A - ib)(a + ib) = A 2 - (ib) 2 = A 2 + B 2 (i.e. real) +
9 Expectation values Expectation value: the most probable value of a variable Multiply variable by probability density ( ψ 2 ) and integrate! eg. expectation value of 1-D position: x = If ψ is normalised then denominator = 1, in which case x = + and for a general variable G(x) the expectation value is G x + xψ 2 dx ( ) = G x + ( )ψ 2 dx + xψ 2 dx ψ 2 dx
10 Free particle wavefunction.think simple wave, rather than wavegroup ψ = Aexp[ i( ωt kx) ] ω ω = 2πν = 2π E h = E k = 2π = 2π p λ h = p ψ = Aexp i E t p x Replacing wave-notation (ω, k) with particle notation (E, p) What is the equation of motion, just as in Newton s Laws, but for quantum particle?
11 Free particle functions Now consider partial differential with x Re-arranged: Or: i.e. if we operate on ψ with iħδ/δx - we get the value of momentum p multiplied by ψ iħδ/δx is the momentum operator ˆp 2 Expect Kinetic Energy operator 2m = 2 2 2m x 2 General Equation: ψ = Aexp iet + ipx = Ae iet e ipx ψ x = Ae iet. ip eipx = ip ψ i x ψ = pψ Operator on ψ = value ψ ˆp i x ψ = pψ
12 1-D Schroedinger Equation Is the equation for energy. Consider: ψ = Aexp i E t p x ψ x = +ip ψ 2 ψ x = p2 2 ψ 2 ψ 2 p2 ψ = 2 x 2 ψ t = i E ψ Eψ = +i ψ t Compare with ordinary (non-relativistic) mechanics
13 1-D Schroedinger Equation developed E = KE + PE = p 2 /2m + U(x,t) multiply across by ψ Eψ = (p 2 /2m)ψ + Uψ and substitute for Eψ and p 2 ψ i ψ t = 2 2 ψ 2m x +Uψ 2 1-D Schroedinger Equation i ψ t = 2 2 ψ 2m x + 2 ψ 2 y + 2 ψ 2 z 2 +Uψ 3-D Schroedinger Equation Patently true for free particle (U=0), also found to be true for constrained particle a basic principle
14 Steady State Simplification When U is not a function of t, get considerable simplification: - The time-independent, or steady state Schroedinger Equation. Recall free particle wavefunction: ψ = Aexp i E t p x ψ = Aexp( i E t) exp i p x ψ = ψ exp( i E t) where ψ = Aexp i p x Fortunately, this separation of time and position dependences is also possible for all wavefunctions when U is indep. of t! substitute this form of ψ into 1D Schroedinger Equation.
15 Steady State Schroedinger Equation i ψ t = 2 2 ψ 2m x + Uψ 2 LHS = i t ( ψ exp ( i E t )) = E ψ exp i E t ( ) 2 RHS = 2 ψ exp i E t 2m x 2 = 2 2m exp i E t E ψ = 2 2m drop the dash & re-write as : ( ( )) + U ( ) 2 ψ x + U ψ exp i E 2 t 2 ψ x + U ψ 2 ( ψ exp( i E t )) ( ) 2 ψ x + 2m 2 2 ( E U )ψ = 0
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