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1Summary of Quantum and Statistical Mechanics

1.1 One Dimensional Schrodinger Equation

In Quantum Mechanics, the state of a particle in one dimension and in presence of apotential U(x, t), is entirely described by a complex wave function (x, t) obeyingthe time dependent Schrodinger equation

ih(x, t)

t= h

2

2m 2(x, t)

x2+U(x, t)(x, t)

where m is the mass of the particle and h is the Planck constant, h, divided by 2 .If we multiply the Schrodinger equation by the complex conjugate wave function(x, t), take the complex conjugate of the Schrodinger equation and multiply by(x, t), and nally subtract both expressions, we nd the so-called continuity equa-tion

|(x, t)|2 t

+x

[ih2m

((x, t)

(x, t)x

(x, t)(x, t)x

)]= 0.

This equation represents the conservation law for the quantity |(x, t)|2 dx and

allows us to interpret |(x, t)|2 as the probability density function to nd the particlein the point x at a time t. The quantity

J(x, t) =ih2m

((x, t)

(x, t)x

(x, t)(x, t)x

)

is the density ux for such probability. The physical interpretation of |(x, t)|2 setssome conditions on (x, t) that has to be chosen as a continuous not multivaluedfunction without singularities. Also, the derivatives of (x, t) have to be continu-ous, with the exception of a moving particle in a potential eld possessing somediscontinuities, as we will see explicitly in the exercises. If the potential does notdepend explicitly on time,U(x, t) =U(x), the time dependence can be separated out

Cini M., Fucito F., Sbragaglia M.: Solved Problems in Quantum and Statistical Mechanics.DOI 10.1007/978-88-470-2315-4 1, c Springer-Verlag Italia 2012

4 1 Summary of Quantum and Statistical Mechanics

from the Schrodinger equation and the solutions, named stationary, satisfy

(x, t) = eih Et(x).

In such a case, the functions = ||2 and J are independent of time. Using theform of (x, t) in the original equation, we end up with the stationary Schrodingerequation [

h2

2md2

dx2+U(x)

](x) = H(x) = E(x).

The operator H is known as the Hamiltonian of the system. Continuous, non mul-tivalued and nite functions which are solutions of this equation exist only for par-ticular values of the parameter E, which has to be identied with the energy of theparticle. The energy values may be continuous (the case of a continuous spectrumfor the Hamiltonian H), discrete (discrete spectrum), or even present a discrete andcontinuous part together. For a discrete spectrum, the associated may be normal-ized to unity

|(x)|2 dx = 1.All the functions corresponding to precise values of the energy are called eigen-functions and are orthogonal. With a continuous spectrum, the condition of or-thonormality may be written using the Dirac delta function

E(x)E (x)dx = (EE ).

The condition of continuity for the wave function and its derivatives is valid even inthe case when the potential energy U(x) is discontinuous. Nevertheless, such con-ditions are not valid when the potential energy becomes innite outside the domainwhere we solve our differential equations. The particle cannot penetrate a region ofthe space where U = + (you can imagine electrons inside a box), and in such aregion we must have = 0. The condition of continuity imposes a vanishing wavefunction where the potential energy barrier is innite and, consequently, the deriva-tives may present discontinuities.

Let U be the minimum of the potential. Since the average value of the energy isE = T +U , and since U > U , we conclude that

E > U

due to the positive value of T , that is the average kinetic energy of the particle. Thisrelation is true for a generic state and, in particular, is still valid for an eigenfunctionof the discrete spectrum. It follows that En > U , with En any of the eigenvalues ofthe discrete spectrum. If we now dene the potential energy in such a way that itvanishes at innity (U() = 0), the discrete spectrum is characterized by all thoseenergy levels E < 0 which represent bound states. In fact, if the particle is in a boundstate, its motion takes place between two points (say x1,x2) so that () = 0.

1.2 One Dimensional Harmonic Oscillator 5

This constraints the normalization condition for the states. In Classical Mechanics,the inaccessible regions where E

6 1 Summary of Quantum and Statistical Mechanics

represents the n-th order Hermite polynomial. The rst Hermite polynomials are

H0( ) = 1 H1( ) = 2

H2( ) = 4 22 H3( ) = 8 312 .Equivalently, we can describe the properties of the harmonic oscillator with thecreation and annihilation operators, a and a, such that [a, a] = 11. In this case, theHamiltonian becomes

H = h(aa+

1211)= h

(n+

1211)

where n = aa is the number operator with the property n |n= n |n . The relationsconnecting the creation and annihilation operators with the position and momentumoperators are

a =

m2h x+

i2mh

p

a =

m2h x i2mh p

x =

h2m (a

+ a)

p = i

mh2 (a

a).The creation and annihilation operators act on a generic eigenstate as step up andstep down operators

a |n=n+1 |n+1 a |n=n |n1

so that

|n= (a)nn!

|0 n|m= nm.

1.3 Variational Method

The variational method is an approximation method used to nd approximateground and excited states. The basis for this method is the variational principlewhich we briey describe now. Let | be a state of an arbitrary quantum systemwith one or many particles normalized such that

N = |= 1.

The energy of the quantum system is a quadratic functional of |

E = |H|

1.3 Variational Method 7

and cannot be lower than the ground state 0; in fact, let us expand the state | inthe eigenfunction basis |n (each one corresponding to the eigenvalue n) of H

|=nn||n

n|n||2 = 1.

Since n 0, we nd that

E = |H|=n|n||2n

n|n||20 = 0.

From this we infer that the energy of the ground state can be found by minimization.Let us start by variating the state |

| |+ |.

The variation of the energy which follows is

E = |H|+ |H|+O( 2).

At the same time, the normalization changes as

N = |+ |+O( 2).

The extremum we are looking for must be represented by functions whose norm is1. This condition can be efciently imposed by introducing a Lagrangian multiplierand minimizing the quantity

Q () = |H11|= (EN).

The variation of Q () must be zero for variations in both and

Q () = |H11|= 0 Q () = ( |1) = 0.

The second equation is the constraint, the rst must be valid for arbitrary variations,leading to

H|= |which is the stationary Schrodinger equation. Multiplying by |, we get the valueof the multiplier, i.e. the energy of the ground state. We remark that the condition ofconstrained minimum, follows from that of unconstrained minimum ( |H|) =0 substituting H with (H 11). This can also be done by making | depend onthe multiplier , whose value is xed imposing

N = ( )|( )= 1.

8 1 Summary of Quantum and Statistical Mechanics

1.4 Angular Momentum

From the denition of the angular momentum in Classical Mechanics

LLL = rrr ppp

with rrr (x,y,z), ppp (px, py, pz), we get its quantum mechanical expression, oncethe vectors rrr, ppp are substituted by their correspondent operators. Once the commu-tation rule between rrr, ppp is known, it is immediate to deduce the commutation rulesof the different components of the angular momentum

[Lx, Ly] = ihLz [Ly, Lz] = ihLx [Lx, Lz] =ihLy.

From the theory of Lie Algebras, we know that a complete set of states is deter-mined from a set of quantum numbers whose number is that of the maximum num-ber of commuting operator we can build starting from the generators (in our caseLx, Ly, Lz). One of these operators is the Casimir operator

L2 = L2x + L2y + L

2z

which is commuting with all the generators of the group, i.e. [L2, Li] = 0, i = x,y,z.The other element of this sub algebra is one of the three generators Lx, Ly, Lz: theconvention is to choose Lz. The quantum states are thus labelled by the quantumnumbers l, m such that

L2 |l,m= h2l(l+1) |l,m Lz |l,m= hm |l,m m l m

L+ |l,m= h

(lm)(l+m+1) |l,m+1L |l,m= h

(l+m)(lm+1) |l,m1

where L = (Lx iLy) are known as raising and lowering operators for the z com-ponent of the angular momentum. Other useful relations are

[Lz, L] =hL [L2, L] = 0 [L+, L] = 2hLzL+L = L2 L2z + hLzLL+ = L2 L2z hLz.

When acting on functions of the spherical polar coordinates, the generators and theCasimir operator take the form

Lx = ih(sin

+ cot cos

)

Ly =ih(cos

cot sin

)

1.4 Angular Momentum 9

Lz =ih

L2 =h2[

1sin

(sin

)+

1sin2

2

2

].

To write the state |l,m in spherical coordinates it is useful to introduce the sphericalharmonics

Yl,m( ,) = , |l,mwhich enjoy the property

Yl,m( ,) = (1)m

(2l+1)4

(lm)!(l+m)!

eimPml (cos) m 0

Yl,m( ,) = Yl,|m|( ,) = (1)|m|Y l,|m|( ,) m < 0where Pml (cos) is the associated Legendre polynomial dened by

Pml (u) = (1u2)m/2dm

dumPl(u) 0 m l

Pl(u) =1

2l l!dl

dul

[(u21)l

]where Pl(u) is the Legendre polynomial of order l. Some explicit expressions forl = 0,1,2 are

Y0,0( ,) =14

Y1,0( ,) =

34

cos Y1,1( ,) =

38

sinei

Y2,0( ,) =

516

(3cos2 1) Y2,1( ,) =

158

sin cosei

Y2,2( ,) =

1532

sin2 e2i .

Given the two angular momentum operators L1, L2, we now want to de