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Lecture 2

Quantum mechanics in one dimension


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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.


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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


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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).


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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


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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.


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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.


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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


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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


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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


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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


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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


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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.


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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


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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).


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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.


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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.


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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)


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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.


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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


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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


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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).


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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 )


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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.


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Quantum mechanics in 1d: bound states

1 Rectangular potential well (continued)

2 -function potential


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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


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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.


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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.


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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.


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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


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Quantum mechanics in 1d: beyond local potentials

1 Kronig-Penney model of a crystal

2 Anderson localization

f


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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


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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


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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


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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


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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


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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


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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!

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