Background Theory
In this chapter, we briefly introduce the quantum confinement for different geometries and the formalism of the Fermi’s Golden Rule. More details can be found in any quantum mechanics book, e.g., Refs. [1,2].
4.1
Quantum Co
finement
4.1.1 Three Dimensional Cartesian Coordinates
The time-dependent Schrödinger equation for a spinless particle of mass m moving under the influence of a three-dimensional potential is
h2 2mr~
2W x; y; z; tð Þ þ ^V x; y; z; tð ÞW x; y; z; tð Þ ¼ ih@W x; y; z; tð Þ
@t ð4:1Þ
where ~r is the Laplacian given by ~ r2¼ @ 2 @x2 þ @2 @y2 þ @2 @z2 ð4:2Þ
The wavefunction of a particle in a time-independent potential can be written as a product of spatial and time components:
W x; y; z; tð Þ ¼ w x; y; zð ÞeiEt
h ð4:3Þ
© The Author(s) 2016
A. Govorov et al., Understanding and Modeling Förster-type Resonance Energy Transfer (FRET), Nanoscience and Nanotechnology,
DOI 10.1007/978-981-287-378-1_4
wherew x; y; zð Þ is the solution to the time-independent Schrödinger equation: h2
2mr~
2
w x; y; zð Þ þ ^V x; y; zð Þw x; y; zð Þ ¼ Ew x; y; zð Þ ð4:4Þ which is of the from ^Hw ¼ Ew: If the potential can be separated into the sum of three independent, one-dimensional terms
V xð ; y; zÞ ¼ V xð Þ þ V yð Þ þ V zð Þ ð4:5Þ we can solve (4.4) by the method of separation of variables. This method consists of separating the three-dimensional Schrödinger equation into three independent one-dimensional Schrödinger equations. Then (4.4), in conjugation with (4.5), can be written as ^Hxþ ^Hyþ ^Hz w x; y; zð Þ ¼ Ew x; y; zð Þ ð4:6Þ where ^Ha is given by Ha¼ h 2 2m @2 @a2 þ Vað Þa ð4:7Þ witha ¼ x; y; z:
As V xð ; y; zÞ separates into three independent terms, we can also write w x; y; zð Þ as a product of three functions, each with a single variable:
w x; y; zð Þ ¼ X xð ÞY yð ÞZ zð Þ ð4:8Þ
Substituting (4.8) into (4.6) and dividing it by X xð ÞY yð ÞZ zð Þ; we obtain h2 2m 1 X d2X dx2 þ Vxð Þx þ h2 2m 1 Y d2Y dy2 þ Vyð Þy þ h2 2m 1 Z d2Z dz2 þ Vzð Þz ¼ E ð4:9Þ Since each expression in the square brackets depends on only one of the vari-ables x, y, or z, and since the sum of these three expressions is equal to a constant energy, E, each expression must then be equal to a constant such that the sum of these three constants is equal to E. For instant, the x-dependent expression is given by h2 2m d2 dx2 þ Vxð Þx X xð Þ ¼ ExX xð Þ ð4:10Þ
Similar equations are also applicable for the y and z coordinates, with
Exþ Eyþ Ez¼ E ð4:11Þ
4.1.2 The Box Potential
We begin with the rectangular box potential, which has no symmetry, and then consider the cubic potential, which displays a great deal of symmetry, since x, y, and z axes are equivalent.
4.1.2.1 The Rectangular Box Potential
Considerfirst the case of a spinless particle of mass m confined in a rectangular box of sides Lx, Lyand Lz:
V xð ; y; zÞ ¼ 0 if 0\x\Lx; 0\y\Ly; 0\z\Lz 1 elsewhere
ð4:12Þ which can be written as V xð ; y; zÞ ¼ Vxð Þ þ Vx yð Þ þ Vy zð Þ; withz
Vxð Þ ¼x
0 if 0\x\Lx
1 elsewhere
ð4:13Þ and the potential Vyð Þ and Vy zð Þ have similar forms.z
The wavefunctionw x; y; zð Þ must vanish at the walls of the box. The solutions for this potential are of the form
X xð Þ ¼ ffiffiffiffiffi 2 Lx r sin nxp Lx x nx¼ 1; 2; 3; . . . ð4:14Þ
and the corresponding energy eigenvalues are
Enx ¼ h2 p2 2mL2 x n2x ð4:15Þ
From these expressions we can write the normalized three-dimensional eigen-functions and their corresponding energies:
wnxnynzðx; y; zÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 8 LxLyLz s sin nxp Lx x sin nyp Ly y sin nzp Lz z ð4:16Þ
Enxnynz ¼ h2 p2 2m n2x L2 x þn 2 y L2 y þn2z L2 z ! ð4:17Þ where nx; ny; nz¼ 1; 2; 3; . . ..
4.1.2.2 The Cubic Box Potential
Similarly to the previous case, we consider the case of a spinless particle of mass m confined in a cubic box of side L.
V xð ; y; zÞ ¼ 0 if 0\x\L; 0\y\L; 0\z\L
1 elsewhere
ð4:18Þ Recalling the results obtained for the rectangular case, (4.16) and (4.17), the eigenfunctions and eigenenergies are:
wnxnynzðx; y; zÞ ¼ ffiffiffiffiffi 8 L3 r sin nxp L x sin nyp L y sin nzp L z ð4:19Þ Enxnynz¼ h2 p2 2mL2 n 2 xþ n 2 yþ n 2 z ð4:20Þ The ground state, nx¼ ny¼ nz¼ 1; has energy
E111¼
3h2p2
2mL2 ð4:21Þ
There is threefirst excited states, corresponding to the three combination of nx,
nyand nz, i.e., nx= 2, ny= 1, nz= 1, or nx= 1, ny= 2, nz= 1 or nx= 1, ny= 1,
nz= 2 whose squares sum to 6. Thefirst excited state has energy
E211¼ E121¼ E112¼
6h2p2
2mL2 ð4:22Þ
Note that each of the first excited states is characterized by different wave-function:w211 has wavelength L along the x-axes and wavelength 2L along the
y-and z-axes, but forw121andw121the shorter wavelength is along the y-axis and the
z-axis, respectively.
Whenever different states have the same energy, this energy level is said to be degenerate. In the case above, thefirst excited level is three-fold degenerate. This system has degenerate levels because of the high degree of symmetry associated with the cubic shape of the box. The degeneracy would be lifted, if the sides of the box were of unequal lengths (rectangular box).
4.1.3 Three Dimensional Spherical Coordinates
In this section, we study the structure of the Schrödinger equation for a particle of mass M moving in a spherically symmetric potential
V rð Þ ¼ V rð Þ ð4:23Þ
which is also known as the central potential.
The time-independent Schrödinger equation for a particle of momentum ihr and the potential vector r is
h2 2Mr 2þ V rð Þ w rð Þ ¼ Ew rð Þ ð4:24Þ The Laplacianr2
r separates into a radial part and an angular partr 2 Xas follows r2¼ r2 rþ 1 h2r2r 2 X¼ 1 r2 @ @r r2 @ @r 1 h2r2^L 2 ¼1 r @2 @r2r 1 h2r2^L 2 ð4:25Þ where ^Lis the orbital angular momentum
^L2 ¼ h2 1 sinh @ @h sinh @ @h þ 1 sin2h @2 @u2 ð4:26Þ In spherical coordinates, the Schrödinger takes the form of
h2 2M 1 r @2 @r2r 1 2Mr2^L 2 þ V rð Þ w rð Þ ¼ Ew rð Þ ð4:27Þ
Thefirst term of this equation can be viewed as the radial kinetic energy
h2 2M 1 r @2 @r2r¼ ^P2 r 2M ð4:28Þ
since the radial momentum operator is given in the Hermitian form ^Pr¼ 1 2 r r ^P þ ^P r r h i ¼ ih @ @r þ 1 r ih1 r @ @rr ð4:29Þ
The second term L2=ð2Mr2Þ can be identified with the rotational kinetic energy
the momentum of inertia with respect to the origin of Mr2. In addition, ^L2commute
with ^Lz and ^Has follows
^H; ^L2
h i
¼ ^H; ^Lz
¼ 0 ð4:30Þ
Thus ^H; ^L2; and ^Lz have common eigenfunctions. The simultaneous
eigen-function of ^L2and ^Lz are given by the spherical harmonics
^L2
Ylmðh; uÞ ¼ l l þ 1ð Þh2Ylmðh; uÞ ð4:31Þ
^LzYlmðh; uÞ ¼ mhYlmðh; uÞ ð4:32Þ
The Hamiltonian in (4.27) is a sum of a radial part and an angular part. Thus, we can look for solutions that are products of a radial part and an angular part
w rð Þ ¼ r j nlmh i ¼ wnlmðr; h; uÞ ¼ Rnlð ÞYr lmðh; uÞ ð4:33Þ
The radial wavefunction Rnlð Þ has to be found. The quantum number n isr
introduced to identify the eigenvalues of ^H:
^H nlmj i ¼ Enjnlmi ð4:34Þ
Substituting (4.34) into (4.27) and using the fact thatwnlmðr; h; uÞ is an eigen-function of ^L2 (4.31), then dividing by Rnlð ÞYr lmðh; uÞ and multiplying by 2Mr2,
we obtain an expression where the radial and angular degrees of freedom are separated into h2 r Rnl @2 @r2ðrRnlÞ þ 2Mr 2ðV rð Þ EÞ þ ^L 2 Ylmðh; uÞ Ylmðh; uÞ " # ¼ 0 h2 r Rnl @2 @r2ðrRnlÞ þ 2Mr 2 V rð Þ E ð Þ þ l lð þ 1Þh 2Y lmðh; uÞ Ylmðh; uÞ ¼ 0 ð4:35Þ h2 r Rnl @2 @r2ðrRnlÞ þ 2Mr 2ðV rð Þ EÞ þ l l þ 1ð Þh2¼ 0
The last expression only depends on r and thefinal expression is thus simplified as h2 2M d2 dr2ðrRnlð Þr Þ þ V rð Þ þ l lð þ 1Þh2 2Mr2 rRnlð Þr ð Þ ¼ EnðrRnlð Þr Þ ð4:36Þ
Note that (4.36) does not depend on the azimuthal quantum number m. Thus, the energy En is 2l lð þ 1Þ—fold degenerate. This is due to the fact that, for a given l,
there are 2lð þ 1Þ different eigenfunctions wnlm (i.e., wnll, wnll þ 1, …, wnl l1,
wnl l), which correspond to the same eigenenergy En. This degeneracy property is
peculiar to the central potentials. Moreover, (4.36) has the structure of a one-dimensional equation for r as follows
h 2 2M d2U nlð Þr dr2 þ Veffð ÞUr nlð Þ ¼ Er nUnlð Þr ð4:37Þ
whose solutions give the energy levels of the system with the wavefunction given by
Unlð Þ ¼ rRr nlð Þr ð4:38Þ
and the potential by
Veffð Þ ¼ V rr ð Þ þ
l lð þ 1Þh2
2Mr2 ð4:39Þ
which is known as the effective or centrifugal potential. Here, V(r) is the central potential and l lð þ 1Þh2=ð2Mr2Þ is a repulsive or centrifugal potential, which is
associated with the orbital angular momentum and tends to repel the particle away from the center. wnlmðr; h; uÞ is finite for all values of r spanning from 0 to ∞.
Thus, if Rnlð Þ is finite, rR0 nlð Þ must vanish at r = 0, i.e.,r
lim
r!0½rRnlð Þr ¼ Unlð Þ ¼ 00 ð4:40Þ
4.1.3.1 Free Particle in Spherical Coordinates
Here, we apply the formalism developed above to study the motion of a free particle of mass M and energy Ek¼ h2k2=ð2MÞ where k is the wave vector k ¼ kj j. The
Hamiltonian H¼ h2r2=ð2MÞ of a free particle is rotational invariant and
commutes with ^L2 and ^Lz. Thus, the radial equation for a free particle is
h2 2M 1 r d2 dr2ðrRklð Þr Þ þ l lð þ 1Þ 2Mr2 Rklð Þ ¼ ERr klð Þr ð4:41Þ
using a change of variableq ¼ kr; we reduce this equation into d2Rlð Þq dq2 þ 2 q dRlð Þq dq þ 1 l lð þ 1Þ q2 Rlð Þ ¼ 0q ð4:42Þ
where Rlð Þ ¼ Rq lð Þ ¼ Rkr klð Þ. This differential equation is known as the sphericalr
Bessel equation. The general solutions to this equation are given by the spherical Bessel functions jlð Þ and the spherical Neumann functions nq lð Þ. Since theq
Neumann functions nlð Þ diverge at the origin, and since the wavefunctionsq
wklmðr; h; uÞ are required to be finite everywhere in space, only the spherical Bessel
functions jlð Þ contribute to the eigenfunctions of the free particleq
wklmðr; h; uÞ ¼ jlð ÞYkr lmðh; uÞ ð4:43Þ
where k¼ ffiffiffiffiffiffiffiffiffiffiffiffi2MEk
p
=h. Note that, since the index k in Ek¼ h2k2=ð2MÞ varies
continuously, the energy spectrum of a free particle is infinitely degenerate. This is because all orientations of k in space correspond to the same energy.
4.1.3.2 The Spherical Well
We consider a particle of mass M confined to the interior of a spherical well with impenetrable walls. In the domain r a; the wavefunction vanishes. In the domain r\a; the potential is zero.
V rð Þ ¼ 0 if r\a 1 if r a
ð4:44Þ To impose the boundary conditionw r ¼ að Þ ¼ 0; we set
jlð Þ ¼ jka lð Þ ¼ jx lð Þ ¼ 0xnl ð4:45Þ
where x ka and xnl is the nth zero of jlð Þ: Thus, the eigenfunctions and eigen-x
values for the spherical well are given by nth wnlmðr; h; uÞ ¼ jl xnl a r Ylmðh; uÞ ð4:46Þ Enl¼ h2x2 nl 2Ma2 ð4:47Þ
4.1.4 Three Dimensional Cylindrical Coordinates
We next consider the case of a particle of mass M confined to a cylindrical box of radius a and length L; that is,
V rð ; /; zÞ ¼ 0 r\a 0\z\L
1 elsewhere
Employing the Hamiltonian in cylindrical coordinates, the Schrödinger equation for the confined particle is given by
@2 w @r2 þ 1 r @w @r þ 1 r2 @2 w @/2þ @2 w @z2 þ k 2 w ¼ 0 ð4:49Þ E¼h 2k2 2M ð4:50Þ
with the separation of coordinates
w r; /; zð Þ ¼ R rð ÞU /ð ÞZ zð Þ ð4:51Þ Eq. (4.49) becomes 1 R @2R @r2 þ 1 r @R @r þ 1 r2 1 U @2U @/2 þ 1 Z @2Z @z2 þ k 2 ¼ 0 ð4:52Þ It follows that 1 Z @2Z @z2 ¼ constant k 2 z ð4:53Þ 1 U @2U @/2¼ constant ¼ m 2 ð4:54Þ 1 R r 2@2R @r2 þ r @R @r þ r2 k2 k2 z ¼ m2 ð4:55Þ
Applying boundary conditions (4.48), wefind
Z zð Þ ¼ A sin kð Þ; kzz zL¼ nzp; nz¼ 1; 2; . . . ð4:56Þ
Furthermore, asU /ð Þ ¼ U / þ 2pð Þ, we obtain
U /ð Þ ¼ Beim/; m ¼ 0; 1; 2; . . . ð4:57Þ
where A and B are constant. Returning to (4.55) and labelling
k2 k2z K2 ð4:58Þ
There results q2d 2R dq2 þ q dR dq þ q 2 m2 R¼ 0 ð4:60Þ
which is known as Bessel’s equation. General solution to this equation are given by
Rð Þ ¼ Cq 1Jmð Þ þ Cq 2Nmð Þq ð4:61Þ
where C1and C2are constants. The functions Jmð Þ and Nq mð Þ are called Besselq
and Neumann functions of thefirst kind, respectively. Since Nmð Þ ¼ 1; the only0
acceptable solution for the wavefunction are Jmð Þ. The remaining boundaryq
conditions gives
R rð ¼ aÞ ¼ C1Jmð Þ ¼ 0aK ð4:62Þ
Let us call the sthfinite zero of Jmð Þ; xq msso that
JmðaKmsÞ Jmð Þ ¼ 0xms ð4:63Þ
Thus, the eigenfunctions are given by wmsnzðr; /; zÞ ¼ AJm xmx a r sin nzp L z eim/ ð4:64Þ
with m≥ 0, s > 0, nz≥ 1, and all three parameters are integers. The corresponding
eigenenergies are E¼h 2k2 2M ¼ h2 2M K 2þ k2 z ¼ h2 2M K 2 msþ nzp L 2 ð4:65Þ with xms¼ aKms, the former becomes
Emsnz¼ h2 2M x2 ms a2 þ nzp L 2 ð4:66Þ
4.2
Fermi
’s Golden Rule
The transition probability corresponding to a transition from an initial unperturbed statej i to another unperturbed state wwi f is
Pifð Þ ¼ t i h Zt 0 wf ^V tð Þ w0 j iei ixfit0dt0 2 ð4:67Þ
In the case where ^V does not depend on time, (4.67) reduces to Pifð Þ ¼ wt ^f Vj iwi 2sin2 12hxfit 1 2hxfi 2 ð4:68Þ
As a function of the time, this transition probability is an oscillating sinusoidal function with a period of 2p=xfi. As a function ofxfi, the transition probability, as
shown in (4.68), has an interference pattern: it is appreciable only nearxfi’ 0 and
decays rapidly asxfimoves away from zero (Fig. 4.1). Hence, for afix t, we have
assumed thatxfiis a continuous variable; that is, we have considered a continuum
offinal states. This means that the transition probability of finding the system in a state wf of energy Ef is the greatest only when xfi’ 0 or equivalently when
Ei’ Ef. The height and the width of the main peak, centered aroundxfi¼ 0; are
proportional to t2 and 1=t, respectively. Therefore, the area under the probability curve is proportional to t. Since most of the area is under the central peak, the transition probability is proportional to t. Therefore, the transition probability grows linearly with time. The central peak becomes narrower and stronger as time increases; this is exactly the property of a delta function. Thus, in the limit t! 1, the transition probability takes the shape of a delta function. Therefore, (4.68) boils down to Pifð Þ ¼t 2pt h ^wf Vj iwi 2 d Ef Ei ð4:69Þ Fig. 4.1 sin2 1 2xfit = xfi 2
versusxfifor afixed value of
t whenxfi¼ Ef Ei
To obtain (4.69), we use the following properties: sin2 1 2xfit 1 2xfi 2 ¼ 2pthd hxfi ð4:70Þ hxfi¼ Ef Ei ð4:71Þ
The transition rate, which is defined as the transition probability per unit time, is given by Cif ¼ Pifð Þt t ¼ 2p h ^wf Vj iwi 2 d Ef Ei ð4:72Þ The delta term d Ef Ei
guarantees the conservation of energy: in the limit t! 1; the transition rate is nonvanishing only between the states of equal energy. Hence, a constant perturbation neither removes energy from the system nor supplies energy to it.
Let us now calculate the total transition rate associated with a transition from an initial statej i into a continuum of final states wwi . Iff q Ef
is the density offinal states—the number of states per unit energy intervals—the number of final states within the energy intervals Ef and Efþ dEf is equal toq Ef
dEf. Then, the total
transition rate Wif can be obtained from (4.72):
Wif ¼ Z Pifð Þt t q Ef dEf ¼ 2p h ^wf Vj iwi 2Z q Ef d Ef Ei dEf ð4:73Þ Wif ¼ 2p h ^wf Vj iwi 2 q Eð Þi ð4:74Þ
This relation is called the Fermi’s Golden Rule. It implies that, in the case of a constant perturbation, if we wait long enough, the total transition rate becomes constant.
References
1. R.L. Liboff, Introductory to quantum mechanics, 4th edn. (Addison-Wesley, San Francisco, 2003)
