Lecture2Quantum mechanics in one dimensionQuantum mechanics in1d:Outline1Unbound statesFree particlePotential stepPotential barrierRectangular potential well2Bound statesRectangular potential well(continued)δ-function potential3Beyond local potentialsKronig-Penney model of a crystalAnderson localizationi ∂tΨ(x,t)=− 2∂2x2mΨ(x,t)For V=0Schr¨o dinger equation describes travelling waves.Ψ(x,t)=A e i(kx−ωt),E(k)= ω(k)= 2k2 2mwhere k=2πλwithλthe wavelength;momentum p= k=hλ.Spectrum is continuous,semi-infinite and,apart from k=0,has two-fold degeneracy(right and left moving particles).i ∂tΨ(x,t)=− 2∂2x2mΨ(x,t)Ψ(x,t)=A e i(kx−ωt)For infinite system,it makes no sense tofix wave function amplitude,A,by normalization of total probability.Instead,fix particleflux:j=−2m(iΨ∗∂xΨ+c.c.)j=|A|2 km=|A|2pmNote that definition of j follows from continuity relation,∂t|Ψ|2=−∇·jThe Fourier transform of a normalized Gaussian wave packet,ψ(x)= 12πα 1/4e ik0x e−x24α.(moving at velocity v= k0/m)is also a Gaussian,ψ(k)= 2απ 1/4e−α(k−k0)2,Although we can localize a wave packet to a region of space,this has been at the expense of having some width in k.For the Gaussian wave packet,∆x = [x − x ]2 1/2≡ x 2 − x 2 1/2=√α,∆k =1√4αi.e.∆x ∆k =12,constant.In fact,as we will see in the next lecture,the Gaussian wavepacket has minimum uncertainty ,∆p ∆x = 2Stationary form of Schr¨o dinger equation,Ψ(x,t)=e−iEt/ ψ(x):− 2∂2x2m+V(x) ψ(x)=Eψ(x)As a linear second order differential equation,we must specify boundary conditions on bothψand its derivative,∂xψ. As|ψ(x)|2represents a probablility density,it must be everywherefinite⇒ψ(x)is alsofinite. Sinceψ(x)isfinite,and E and V(x)are presumedfinite,so∂2xψ(x)must befinite.⇒bothψ(x)and∂xψ(x)are continuous functions of xFor E >V 0,both k <and k >=2m (E −V 0)are real,and j i = k <m,j r =|r |2 k <m,j t =|t |2 k >mDefining reflectivity,R ,and transmittivity,T ,R =reflected flux incident flux,T =transmitted flux incident flux R =|r |2=k <−k >k <+k >2,T =|t |2k >k <=4k <k >(k <+k >)2,R +T =1For E <V 0, k >=2m (E −V 0)becomes pure imaginary,wavefunction,ψ>(x ) te −|k >|x ,decays evanescently,andj i = k <m,j r =|r |2 k <m,j t =0Beam is completely reflected from barrier,R =|r |2= k <−k >k <+k >2=1,T =0,R +T =1Transmission across a potential barrier–prototype for generic quantum scattering problem dealt with later in the course. Problem provides platform to explore a phenomenon peculiar to quantum mechanics–quantum tunneling.Wavefunction parameterization:ψ1(x)=e ik1x+r e−ik1x x≤0ψ2(x)=A e ik2x+B e−ik2x0≤x≤aψ3(x)=t e ik1x a≤xwhere k1=√2mE and k2= 2m(E−V0).Continuity conditions onψand∂xψat x=0and x=a,1+r=A+BAe ik2a+Be−ik2a=te ik1a, k1(1−r)=k2(A−B)k2(Ae ik2a−Be−ik2a)=k1te ik1aSolving for transmission amplitude,t=2k1k2e−ik1a2k1k2cos(k2a)−i(k21+k22)sin(k2a)which translates to a transmissivity ofT=|t|2=11+14 k1k2−k2k1 2sin2(k2a) and reflectivity,R=1−T(particle conservation).but penetrates,barrier region–quantumUnbound particles:tunnelingAlthough tunneling is a robust,if uniquely quantum,phenomenon,it is often difficult to discriminate from thermal activation.Experimental realization provided by Scanning TunnelingMicroscope(STM)Quantum mechanical scattering in three-dimensions In three dimensions,plane wave can be decomposed intosuperposition of incoming and outgoing spherical waves: If V(r)short-ranged,scattering wavefunction takes asymptotic form,e i k·r=i2k∞ =0i (2 +1) e−i(kr− π/2)r−S (k)e i(kr− π/2)r P (cosθ)Quantum mechanics in1d:bound states1Rectangular potential well(continued)2δ-function potentialFor a potential well,we seek bound state solutions with energies lying in the range−V0<E<0.Symmetry of potential⇒states separate into those symmetric and those antisymmetric under parity transformation,x→−x. Outside well,(bound state)solutions have form√−2mE>0ψ1(x)=Ceκx for x>a, κ=In central well region,general solution of the formψ2(x)=A cos(kx)or B sin(kx), k= 2m(E+V0)>0Uncertainty relation,∆p∆x>h,shows that confinement by potential well is balance between narrowing spatial extent ofψwhile keeping momenta low enough not to allow escape.In fact,one may show(exercise!)that,in one dimension,arbitrarily weak binding always leads to development of at least one bound state.In higher dimension,potential has to reach critical strength to bind a particle.Forδ-function potential V(x)=−aV0δ(x),− 2∂2x2m−aV0δ(x) ψ(x)=Eψ(x)(Once again)symmetry of potential shows that stationary solutions of Schr¨o dinger equation are eigenstates of parity,x→−x.States with odd parity haveψ(0)=0,i.e.insensitive to potential.Quantum mechanics in1d:beyond local potentials1Kronig-Penney model of a crystal2Anderson localizationKronig-Penney model provides caricature of(one-dimensional) crystal lattice potential,∞ n=−∞δ(x−na)V(x)=aV0Since potential is repulsive,all states have energy E>0. Symmetry:translation by lattice spacing a,V(x+a)=V(x). Probability density must exhibit same translational symmetry, |ψ(x+a)|2=|ψ(x)|2,i.e.ψ(x+a)=e iφψ(x).In region(n−1)a<x<na,general solution of Schr¨o dinger equation is plane wave like,ψn(x)=A n sin[k(x−na)]+B n cos[k(x−na)]√2mEwith k=Imposing boundary conditions onψn(x)and∂xψn(x)and requiring ψ(x+a)=e iφψ(x),we can derive a constraint on allowed k values (and therefore E)similar to quantized energies for bound states.Rearranging equations(1)and(2),and using the relations A n+1=e iφA n and B n+1=e iφB n,we obtaincosφ=cos(ka)+maV02k sin(ka)Since cosφcan only take on values between−1and1,there are2k2Example:Naturally occuring photonic crystals “Band gap”phenomena apply to any wave-like motion in a periodicsystem including light traversing dielectric media,e.g.photonic crystal structures in beetles and butterflies!Band-gaps lead to perfect reflection of certain frequencies.Anderson localizationWe have seen that even a weak potential can lead to the formationof a bound state.However,for such a confining potential,we expect high energystates to remain unbound.Curiously,and counter-intuitively,in1d a weak extended disorderpotential always leads to the exponential localization of allquantum states,no matter how high the energy!First theoretical insight into the mechanism of localization wasachieved by Neville Mott!。
