lec2 compressed
TRANSCRIPT
-
7/30/2019 Lec2 Compressed
1/38
Lecture 2
Quantum mechanics in one dimension
-
7/30/2019 Lec2 Compressed
2/38
Quantum mechanics in one dimension
Schrodinger equation for non-relativistic quantum particle:
it(r, t) = H(r, t)
whereH =
2
2
2m + V(r) denotes quantum Hamiltonian.
To acquire intuition into general properties, we will review somesimple and familiar(?) applications to one-dimensional systems.
Divide consideration between potentials, V(x), which leave particlefree (i.e. unbound), and those that bind particle.
-
7/30/2019 Lec2 Compressed
3/38
Quantum mechanics in 1d: Outline
1 Unbound statesFree particlePotential stepPotential barrierRectangular potential well
2 Bound statesRectangular potential well (continued)-function potential
3 Beyond local potentials
Kronig-Penney model of a crystalAnderson localization
-
7/30/2019 Lec2 Compressed
4/38
Unbound particles: free particle
it(x, t) =
22
x
2m
(x, t)
For V = 0 Schrodinger equation describes travelling waves.
(x, t) = A ei(kxt), E(k) = (k) =
2k2
2m
where k = 2 with the wavelength; momentum p = k =h .
Spectrum is continuous, semi-infinite and, apart from k = 0, hastwo-fold degeneracy (right and left moving particles).
-
7/30/2019 Lec2 Compressed
5/38
Unbound particles: free particle
it(x, t) = 22
x
2m(x, t) (x, t) = A ei(kxt)
For infinite system, it makes no sense to fix wave functionamplitude, A, by normalization of total probability.
Instead, fix particle flux: j = 2m
(ix + c.c.)
j = |A|2k
m
= |A|2p
mNote that definition of j follows from continuity relation,
t||2 = j
-
7/30/2019 Lec2 Compressed
6/38
Preparing a wave packet
To prepare a localized wave packet, we can superpose componentsof different wave number (cf. Fourier expansion),
(x) =12
(k) eikxdk
where Fourier elements set by
(k) =12
(x) eikxdx.
Normalization of(k) follows from that of(x):
(k)(k)dk =
(x)(x)dx = 1
Both |(x)|2dx and |(k)|2dk represent probabilities densities.
-
7/30/2019 Lec2 Compressed
7/38
Preparing a wave packet: example
The Fourier transform of a normalized Gaussian wave packet,
(x) =
1
2
1/4eik0xe
x2
4 .
(moving at velocity v = k0/m) is also a Gaussian,
(k) = 2 1/4
e(kk0)
2
,
Although we can localize a wave packet to a region of space, thishas been at the expense of having some width in k.
-
7/30/2019 Lec2 Compressed
8/38
Preparing a wave packet: example
For the Gaussian wave packet,
x =
[x x]21/2 x2 x21/2 = , k = 14
i.e. xk =1
2, constant.
In fact, as we will see in the next lecture, the Gaussian wavepacket
has minimum uncertainty,
px =
2
-
7/30/2019 Lec2 Compressed
9/38
Unbound particles: potential step
Stationary form of Schrodinger equation, (x, t) = eiEt/(x):
22x
2m+ V(x)
(x) = E(x)
As a linear second order differential equation, we must specifyboundary conditions on both and its derivative, x.
As |(x)|2 represents a probablility density,it must be everywhere finite (x) is also finite.
Since (x) is finite, and E and V(x) are presumed finite,so 2
x(x) must be finite.
both x and x x are continuous functions of x
-
7/30/2019 Lec2 Compressed
10/38
Unbound particles: potential step
22x
2m+ V(x)
(x) = E(x)
Consider beam of particles (energy E) moving from left to rightincident on potential step of height V0 at position x = 0.
If beam has unit amplitude, reflected and transmitted (complex)amplitudes set by r and t,
0
where k< =
2mE and k> = 2m(EV0).
Applying continuity conditions on and x at x = 0,
(a) 1 + r = t(b) ikt r =
k< k>k< + k>
, t =2k
-
7/30/2019 Lec2 Compressed
11/38
Unbound particles: potential step
For E > V0, both k< and k> = 2m(E V0) are real, and
ji =kk< + k>
2, T = |t|2
k>
k)2
, R + T = 1
-
7/30/2019 Lec2 Compressed
12/38
Unbound particles: potential step
For E < V0, k> = 2m(E V0) becomes pure imaginary,wavefunction, >(x) te|k>|x, decays evanescently, andji =
k
2
= 1, T = 0, R + T = 1
-
7/30/2019 Lec2 Compressed
13/38
Unbound particles: potential barrier
Transmission across a potential barrier prototype for genericquantum scattering problem dealt with later in the course.
Problem provides platform to explore a phenomenon peculiar to
quantum mechanics quantum tunneling.
-
7/30/2019 Lec2 Compressed
14/38
Unbound particles: potential barrier
Wavefunction parameterization:
1(
x) =
eik1x
+r eik1x x
02(x) = A eik2x + B eik2x 0 x a3(x) = t e
ik1x a x
where k1 =
2mE and k2 =
2m(E V0).Continuity conditions on and x at x = 0 and x = a,
1 + r = A + BAeik2a + Beik2a = teik1a
,
k1(1 r) = k2(A B)k2(Ae
ik2a Beik2a) = k1teik1a
-
7/30/2019 Lec2 Compressed
15/38
Unbound particles: potential barrier
Solving for transmission amplitude,
t =
2k1k2eik1a
2k1k2 cos(k2a) i(k21 + k22 ) sin(k2a)which translates to a transmissivity of
T = |t|2 =1
1 + 14 k1k2 k2k12
sin2(k2a)
and reflectivity, R = 1 T (particle conservation).
-
7/30/2019 Lec2 Compressed
16/38
Unbound particles: potential barrier
T = |t|2 = 1
1 + 14
k1
k2 k2
k1
2sin2(k2a)
For E > V0 > 0, T shows oscillatory behaviour with T reachingunity when k2a
a
2m(EV0) = n with n integer.
At k2a = n, fulfil resonance condition: interference eliminatesaltogether the reflected component of wave.
-
7/30/2019 Lec2 Compressed
17/38
Unbound particles: potential barrier
T = |t|2 = 1
1 + 14
k1
k2 k2
k1
2sin2(k2a)
For V0 > E > 0, k2 = i2 turns pure imaginary, and wavefunctiondecays within, but penetrates, barrier region quantum tunneling.
For 2a 1 (weak tunneling), T 16k21
22
(k21 + 22)
2e22a.
-
7/30/2019 Lec2 Compressed
18/38
Unbound particles: tunneling
Although tunneling is a robust, if uniquely quantum, phenomenon,it is often difficult to discriminate from thermal activation.
Experimental realization provided by Scanning TunnelingMicroscope (STM)
-
7/30/2019 Lec2 Compressed
19/38
Unbound particles: potential well
T = |t|2 =1
1 + 14
k1k2 k2
k1
2 sin2(k2a)
For scattering from potential well (V0 < 0), while E > 0, result stillapplies continuum of unbound states with resonance behaviour.
However, now we can find bound states of the potential well withE < 0.
But, before exploring these bound states, let us consider the generalscattering problem in one-dimension.
-
7/30/2019 Lec2 Compressed
20/38
Quantum mechanical scattering in one-dimension
V(x)Aeikx
Beikx Ceikx
Deikx
Consider localized potential, V(x), subject to beam of quantum
particles incident from left and right.Outside potential, wavefunction is plane wave with k =
2mE.
Relation between the incoming and outgoing components of planewave specified by scattering matrix (or S-matrix)
CB
= S11 S12S21 S22
AD
= out = Sin
-
7/30/2019 Lec2 Compressed
21/38
Quantum mechanical scattering in one-dimension
V(x)Aeikx
Beikx Ceikx
Deikx
With jleft =k
m(|A|2 |B|2) and jright = km (|C|2 |D|2), particle
conservation demands that jleft = jright, i.e.
|A|2 + |D|2 = |B|2 + |C|2 or inin = outout
Then, since out = Sin,
inin
!= outout =
in S
S!= I
in
and it follows that S-matrix is unitary: SS = I
-
7/30/2019 Lec2 Compressed
22/38
Quantum mechanical scattering in one-dimension
V(x)Aeikx
Beikx Ceikx
Deikx
For matrices that are unitary, eigenvalues have unit magnitude.
Proof: For eigenvector |v, such that S|v = |v,v|SS|v = ||2v|v = v|v
i.e. ||2 = 1, and = ei.
S-matrix characterised by two scattering phase shifts,e2i1 and e2i2 , (generally functions of k).
-
7/30/2019 Lec2 Compressed
23/38
Quantum mechanical scattering in three-dimensions
In three dimensions, plane wave can be decomposed into
superposition of incoming and outgoing spherical waves:
If V(r) short-ranged, scattering wavefunction takes asymptoticform,
eikr =i
2k
=
i(2 + 1)
ei(kr/2)
r S(k) e
i(kr/2)
r
P(cos )
-
7/30/2019 Lec2 Compressed
24/38
Quantum mechanical scattering in one-dimension
V(x)Aeikx
Beikx Ceikx
Deikx
For a symmetric potential, V(x) = V(x), S-matrix has the form
S = t rr t where r and t are complex reflection and transmission amplitudes.
From the unitarity condition, it follows that
SS = I = |t|2 + |r|2 rt + rt
rt
+ r
t |t|2 + |r|2 i.e. rt + rt = 0 and |r|2 + |t|2 = 1 (or r2 = t
t(1 |t|2)).
For application to a -function potential, see problem set I.
-
7/30/2019 Lec2 Compressed
25/38
Quantum mechanics in 1d: bound states
1 Rectangular potential well (continued)
2 -function potential
-
7/30/2019 Lec2 Compressed
26/38
Bound particles: potential well
For 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 andthose antisymmetric under parity transformation, x x.Outside well, (bound state) solutions have form
1(x) = Cex for x > a, =
2mE > 0
In central well region, general solution of the form
2(x) = A cos(kx) or Bsin(kx), k =
2m(E + V0) > 0
-
7/30/2019 Lec2 Compressed
27/38
Bound particles: potential well
Applied to even states,1(x) = Ce
x, 2(x) = A cos(kx),
continuity of and x implies
Cea = A cos(ka)
Cea = Aksin(ka)
(similarly odd).
Quantization condition:
a =
ka tan(ka) evenka cot(ka) odd
a =2ma2V0
2 (ka)2
1/2
at least one bound state.
-
7/30/2019 Lec2 Compressed
28/38
Bound particles: potential well
Uncertainty relation, px > h, shows that confinement bypotential well is balance between narrowing spatial extent of whilekeeping momenta low enough not to allow escape.
In fact, one may show (exercise!) that, in one dimension, arbitrarilyweak binding always leads to development of at least onebound state.
In higher dimension, potential has to reach critical strength to binda particle.
-
7/30/2019 Lec2 Compressed
29/38
Bound particles: -function potential
For -function potential V(x) = aV0(x),
22x
2m aV0(x)
(x) = E(x)
(Once again) symmetry of potential shows that stationary solutions
of Schrodinger equation are eigenstates of parity, x x.States with odd parity have (0) = 0, i.e. insensitive to potential.
-
7/30/2019 Lec2 Compressed
30/38
Bound particles: -function potential
22x
2m aV0(x)
(x) = E(x)
Bound state with even parity of the form,
(x) = A
ex x < 0ex x > 0
, =2mE
Integrating Schrodinger equation across infinitesimal interval,
x|+ x| = 2maV02
(0)
find =maV0
2, leading to bound state energy E = ma
2V2022
-
7/30/2019 Lec2 Compressed
31/38
Quantum mechanics in 1d: beyond local potentials
1 Kronig-Penney model of a crystal
2 Anderson localization
f
-
7/30/2019 Lec2 Compressed
32/38
Kronig-Penney model of a crystal
Kronig-Penney model provides caricature of (one-dimensional)
crystal lattice potential,
V(x) = aV0
n=
(x na)
Since 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,|(x + a)|2 = |(x)|2, i.e. (x + a) = ei(x).
K i P d l f l
-
7/30/2019 Lec2 Compressed
33/38
Kronig-Penney model of a crystal
In region (n
1)a < x < na, general solution of Schrodinger
equation is plane wave like,
n(x) = An sin[k(x na)] + Bn cos[k(x na)]
with k =
2mE
Imposing boundary conditions on n(x) and xn(x) and requiring(x + a) = ei(x), we can derive a constraint on allowed k values(and therefore E) similar to quantized energies for bound states.
K i P d l f l
-
7/30/2019 Lec2 Compressed
34/38
Kronig-Penney model of a crystal
n(x) = An sin[k(x na)] + Bn cos[k(x na)]
Continuity of wavefunction, n(na) = n+1(na), translates to
Bn+1
cos(ka) = Bn + An+1
sin(ka) (1)
Discontinuity in first derivative,
xn+1|x=na xn|na = 2maV02
n(na)
leads to the condition,
k[An+1 cos(ka) + Bn+1 sin(ka) An] = 2maV02
Bn (2)
K i P d l f l
-
7/30/2019 Lec2 Compressed
35/38
Kronig-Penney model of a crystal
Rearranging equations (1) and (2), and using the relations
An+1 = eiAn and Bn+1 = eiBn, we obtain
cos = cos(ka) +maV0
2ksin(ka)
Since cos can only take on values between 1 and 1, there are2 2
E l N ll i h i l
-
7/30/2019 Lec2 Compressed
36/38
Example: Naturally occuring photonic crystals
Band gap phenomena apply to any wave-like motion in a periodic
system including light traversing dielectric media,
e.g. photonic crystal structures in beetles and butterflies!
Band-gaps lead to perfect reflection of certain frequencies.
A d l li ti
-
7/30/2019 Lec2 Compressed
37/38
Anderson localization
We have seen that even a weak potential can lead to the formation
of a bound state.
However, for such a confining potential, we expect high energystates to remain unbound.
Curiously, and counter-intuitively, in 1d a weak extended disorderpotential always leads to the exponential localization of all
quantum states, no matter how high the energy!
First theoretical insight into the mechanism of localization wasachieved by Neville Mott!
S Q t h i i 1d
-
7/30/2019 Lec2 Compressed
38/38
Summary: Quantum mechanics in 1d
In one-dimensional quantum mechanics, an arbitrarily weakbinding potential leads to the development of at least one
bound state.For quantum particles incident on a spatially localized potentialbarrier, the scattering properties are defined by a unitary S-matrix,
out = Sin .
The scattering properties are characterised by eigenvalues of the
S-matrix, e2ii.
For potentials in which E < Vmax, particle transfer across thebarrier is mediated by tunneling.
For an extended periodic potential (e.g. Kronig-Penney model), thespectrum of allow energies show band gaps where propagatingsolutions dont exist.
For an extended random potential (however weak), all states arelocalized, however high is the energy!