Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev

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Physics : Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev

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Consider a spinless particle of mass m in a central potential V(r).  The potential energy of the particle depends only on its distance from the origin.

For a classical particle in a central potential the force  Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev is always directed towards the origin, the torque t=r´F  is zero, and the angular momentum L=r´p is a constant of motion.  The particle's trajectory lies in a plane passing through the origin.  Its velocity v may be decomposed into a component vr along Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev and a component v^ perpendicular to Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev in this plane.  The energy of the particle is given by

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

Using p=mv and L=mrv^ we can write

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev

with

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

To make the transition to quantum mechanics we write down the quantum-mechanical Hamiltonian H of the particle by replacing in the expression for E the classical physical quantities expressed in terms of r and p by their corresponding quantum mechanical operators, symmetrizing suitably, if necessary.  In coordinate representations we have already found expressions for the operators px, py, pz, p2, and L2, but we have not yet found an expression for pr2.

In coordinate representation we write

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev

The expression for pr2 in coordinate representation therefore is

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

To find the wave functions y(r,q,f) of the eigenstates of H we have to solve the eigenvalue equation Hy(r)=Ey(r).  For a particle in a central potential the entire angular dependence of H is contained in the L2 term.  L2 commutes with all components Li of L.  All other terms of H depend only on r, not on q and f, and commute with Li.  We have [H,Li]=0, [H,L2]=0.  The angular momentum L of the particle is therefore a constant of motion.  We can find a common eigenbasis of H, L2 and Lz.  We denote the states of this basis by |k,l,m> and the corresponding eigenfunctions by yklm(r,q,f).  We have

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

We already know the general form of yklm(r,q,f).  yklm(r,q,f)=Rkl(r)Ylm(q,f) is a product of a radial function Rkl(r) and the spherical harmonic Ylm(q,f).  (See notes!)

Substituting this form of yklm(r,q,f) into the eigenvalue equation yields the differential equation

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev

for Rkl(r).

Note: Since H commutes with L±E cannot depend on m.  E is therefore at least (2l+1) fold degenerate.  Accidental degeneracies may add to this essential degeneracy.

Writing Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev we find a differential equation for ukl(r), which has a simpler form than the differential equation for Rkl(r).

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

This differential equation has the same form as the one-dimensional Schroedinger equation for a particle moving in an effective potential

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

(Here is one reason why we have spend so much time on 1-d problems.)

But r is not x.  The variable r can take on only non-negative real values.  R(r) has to stay finite at r=0, therefore u(r) has to go to zero at r=0.  Otherwise y(r,q,f) is not an acceptable wave function.  Choosing Veff(r)=¥ for r<0 lets us treat the radial equation as an ordinary one-dimensional equation while guaranteeing the proper behavior at the origin.

 

The asymptotic behavior of Rkl(r)

Assume Rkl(r) is sufficiently regular that it may be expanded in powers of r near the origin.

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev,

where s is the lowest power with a non-zero coefficient.

Therefore

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

Substituting this expression for ukl(r) into

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev

yields

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev

+ (terms containing powers of r greater than s-1) = 0, if |V(r)|£ C'r-2 as r®0.

The coefficients of successive powers of r have to vanish independently, since this equation holds for all r.  Letting the coefficient of rs-1 vanish we obtain -s(s+1)+l(l+1)=0 if |V(r)|< C'r-2 as r®0, i.e. if |V(r)| does not blow up as fast as r-2 as r®0.  This is a quadratic equation with two solutions.  It implies that s=l, or that s=-(l+1) if |V(r)|< r-2 as r®0.  As r®0 the lowest power of r dominates and ukl(r)®Crl+1.  The solution ukl(r)®Crl+1 goes to zero at r=0, but the solution ukl(r)®Cr-l blows up at r=0 and is unacceptable.

Near the origin the radial behavior of an acceptable wave function of a particle moving in a central potential is proportional to rl, (if |V(r)|<C'r-2). Note: R(r)=u(r)/r.

 

Two interacting particles

Consider an isolated system of two interacting spinless particles of masses m1 and m2 at positions r1 and r2, respectively.  The potential energy of the particles depends only on their relative position r1-r2.  In classical mechanics the Lagrangian of such a particle can be written as

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev,

where M=m1+m2 is the total mass of the system, 

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev is the reduced mass of the system, 

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev are the coordinates of the center of mass of the system, 

and r=r1-r2 are the relative coordinates.  

Using  Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRevwe define  Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.  The classical equations of motion are obtained from

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

They are

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

If we take PR and pr as the new canonical variables of the system then the Hamiltonian of the system is

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

The first term represents the kinetic energy associated with the motion of the center of mass, which is in uniform rectilinear motion ( P = 0).  The other terms represent the energy associated with the relative motion.  If we choose an inertial frame in which the center of mass is at rest, then  

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev

is the Hamiltonian of a fictitious particle with mass m, whose position is given by the relative coordinate r, moving in an external potential V(r).  The relative motion of two interacting spinless particles reduces to the motion of a single fictitious particle in an external potential.

To make the transition to quantum mechanics we again replace in the expression for H the classical physical quantities expressed in terms of the canonical variables by their corresponding quantum mechanical operators and symmetrize suitably, if necessary.

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev,

where H, Pp, and r are now operators.  Expressing these operators in terms of p1p2r1, and r2 and using Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev we can show that  Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev  for all i and j.

We therefore have

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

There exists a common eigenbasis of H, HCM, and Hr.  We may consider the state space E to be the tensor product of ECM and Er, E=ECMÄEr.  P and R operate only in ECM and p and r operate only in Er.  The motion of the center of mass and the relative motion are completely independent of each other.

Let {|c>} be an eigenbasis of HCM and {|w>} be an eigenbasis of Hr.  Then {|f>=|c>Ä|w>}is an eigenbasis of H in E.

 ,Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev .

In coordinate representation we write

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev

We can solve for c(R).

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

The fictitious particle associated with the motion of the center of mass behaves like a free particle and is represented by a plane wave.

We are often only interested in the relative motion, i.e. in the behavior of the two interacting particles in their center of mass frame.  If their mutual interaction depends only on the distance between them, then V(r)=V(r) and the eigenvalue equation becomes

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

Then

.Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRevis the radial function of a particle of mass m moving in a central potential.


Problem:

Assume that the potential energy of the deuteron is given by V(r)=-V0, r<r0; V(r)=0 r³r0.

(a) Show that the ground state of the deuteron possesses zero orbital angular momentum (l=0).  Since this is true for any central potential, you may not need the detailed nature of the square well potential.

(b) Assume that l=0 and estimate the value of V0 under the additional condition that the value of the binding energy is much smaller than V0.

 

Solution:

(a)  The Hamiltonian of the relative motion of the two particles is

 

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev

 

Let yl be the lowest energy eigenfunction of H with orbital angular momentum quantum number l.

 

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev .

 

Let yl+1 be the lowest energy eigenfunction of H with orbital angular momentum quantum number l+1.

 

 Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev

 

since yl is assumed to be the lowest energy eigenfunction of the Hamiltonian Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

Therefore Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev and the ground state possesses zero orbital angular momentum.

(b)  The relative motion of the two particles is described in the same way as the motion of a fictitious particle of reduced mass m in a central potential.  Therefore we have

 Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

The problem is reduced to a one dimensional "square well problem" with E<0.

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev

Let

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

Let r<r0 define region 1 and r>r0 define region 2.  The coordinate r is never negative.  Therefore

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

The boundary conditions are

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev

in regions where Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

If we plot Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRevwe find solutions at the intersections of the two curves in regions where Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev, i.e. Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

For only one solution to exist we need Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev.

Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev

(Rule of thumb: To estimate the range of a force, divide Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev by the mass of the particle that carries it times c, Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics Physics Notes | EduRev).

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