which implies that we can take solutions which are simultaneous eigen functions of

Module 1 : Quantum Mechanics Chapter 6 : Quantum mechanics in 3-D Quantum mechanics in 3-D For most physical systems, the dynamics is in 3-D. The solutions to the general 3-d problem are quite complicated, and simplify mainly in the case where the Hamiltonian operator separates into sums of terms which depend only on one variable each. One of the most important physical cases is that in which the potential is spherically symmetric, a central potential. In this case the angular momentum plays a very important role. With the one-particle Hamiltonian (6.1) one has (6.2) It will be shown that which implies that we can take solutions which are simultaneous eigen functions of. 6.1 Central potential, angular momentum The Schroedinger equation for the Hamiltonian with central potential, is (6.3) For the appropriate spherical coordinates, one has (6.4) for the metric elements defined in Eq.(5.1). Using Eq.(5.5) and Eq.(5.9), we get for the gradient and Laplacian operators, (6.5) with being unit vectors parallel to resulting from changes in spherical coordinates. The corresponding Schroedinger equation is

(6.6) For the special, separable solutions, use of (6.7) in the Schroedinger equation leads to (6.8) where in the first two equations is the quantum number, and is the mass in the third equation. The first equation may be written as (6.9) which corresponds to eigen function of in Eq.(5.12) with eigenvalue. Also the second equation may be written as (6.10) which is the square of the total angular momentum operator. One can deduce this relation directly. With, and the gradient in Eq.(6.5), one obtains (6.11) Substituting

(6.12) one gets (6.13) This implies (6.14) which leads to the same expression as in Eq.(6.10). There are some important and useful commutation relations for the angular momentum operator. Specifically, one has (6.15) This can be generalised to (6.16) where take the values corresponding to components, and is the antisymmetric tensor of rank, and the repeated index is to be summed over. This also leads to (6.17) Finally, for the operator, one has (6.18) 6.2 Solutions for the angular part For the part in Eq.(6.8), one has

(6.19) where the integer value of is implied by the condition that the wave function is the same for and. This solution is the eigenfunction of in Eq.(6.13) with eigenvalue. For the -part in Eq.(6.8), we take which leads to (6.20) where we have replaced by. This is the generalised Legendre equation with associated Legendre functions as solutions. For the special case of, the equation is the Legendre equation with Legendre polynomials as the solutions, (6.21) Here also, since the equation does not change with, we consider solutions of the form (6.22) Substituting this in Eq.(6.21) leads to (6.23) Taking the coefficient of we obtain (6.24) Since, we get

(6.25) for even or odd solutions. Now for, one has which implies that the solution diverges at. For obtaining convergent solutions, one imposes the condition (6.26) where we have taken to be positive. With this, we can write the the relation in Eq.(6.24) in the form (6.27) with highest power term being. These solutions are the Legendre polynomials. A very convenient and useful expression for the Legendre polynomials is in the form of Rodrigues' formula, (6.28) where represents a derivative with respect to. To show that it satisfies the Legendre equation, we consider (6.29) Operating by, one obtains (6.30) This leads to (6.31) which implies that Rodrigues' expression in Eq.(6.28) satisfies the Legendre equation in Eq.(6.21). Rodrigues' formula in Eq.(6.28) leads to

(6.32) For, the solutions to Eq.(6.20) are given by (6.33) with representing derivative with respect to. To show this, we take number of derivatives of Eq.(6,21) satisfied by, leading to (6.34) With the substitution of (6.35) one obtains (6.36) which is the same as Eq.(6.20). This implies that (6.37) is the solution to the generalised Legendre equation in Eq.(6.20). The combined angular solution is given in the form of spherical harmonics, (6.38) 6.3 Some properties of and functions Some important properties of and are the following: (1) The value of at :

(6.39) (2) Normalization of : (6.40) For, one has (6.41) This integral is related to the -function: (6.42) Together, Eqs(6.40),(6.41),(6.42) imply (6.43) (3) Normalization of : The factor of and the integration over imply that are orthogonal for. We consider the case of,, (6.44)

For, we get (6.45) where we have used the integral from Eq.(6.42). Together, we can write (6.46) (4) Generating function for : Consider the solution for with for the angular part. Then the radial part of the wave function in Eq.(6.8) with, satisfies the equation (6.47) Since the two terms in this equation have the same dimension, we consider a power solution which when substituted in Eq.(6.47) leads to (6.48) For the non-singular solutions, we consider only the solution. Now since (6.49) we consider a general solution in the form (6.50) where we have taken along the direction. Taking,, we use the result in Eq.(6.39) to obtain

(6.51) This can be written in the form (6.52) where is the angle between and, is the smaller and is the larger of and. Finally, taking, one obtains from Eq.(6.51) (6.53) This is regarded as the generating function of Legendre polynomials in terms of the coefficients of. Expanding the function on the left hand side in powers of, we get (6.54) We can also use the generating function to obtain some useful recursion relations. Taking the derivative of Eq.(6.53) with respect to, one gets (6.55) Equating the coefficients of leads to (6.56) which allows us to obtain successive Legendre polynomials. 6.4 Angular momentum algebra One can use the algebra of angular momentum operators to generate various eigenstates of angular momentum. We start with some of the relations specified in Eqs.(6.17) and (6.18), (6.57) The first commutator equation implies that we have a complete set of simultaneous eigenstates of and. The spherical harmonics are the eigenfunctions corresponding to these states,

(6.58) Now we use the algebra to relate different states, starting with : (6.59) which implies that the operation by leads to a state with the eigenvalue of increased by one unit. To deduce the constant, we take the norm of the two sides in Eq.(6.59) which leads to (6.60) Now, we consider, (6.61) Together we have (6.62) which implies that and are raising and lowering operators of eigenstates of. Now the process of raising and lowering by has to terminate, otherwise the condition will be violated. Let and be the maximum and minimum values of for a given. This implies

(6.63) Here, and are left out since they violate the condition. Now since one must reach from in integer number of steps, one has (6.64) For spatial dependent wave functions, we obtain from Eq.(6.13), (6.65) and the eigenfunctions in the form of spherical harmonics in Eq.(6.38). It may also be noted that since the wave function has to be the same for and, and are integers. The half integer values of and consistent with Eq.(6.64), are associated with intrinsic properties, not spatial, and are described as intrinsic spin properties. 6.5 Free particle in 3-D For a free particle in 3-D, the Schroedinger equation in cartesian coordinates is (6.66) For separable solutions, we get (6.67) The solutions can be written in the form (6.68) Now we consider the wave function in terms of spherical coordinates and the complete set of solutions in terms of Legendre polynomials, (6.69) where is the angle between and, and is the solution to the radial part of the Schroedinger equation in Eq.(6.8) with. Multiplying this equation by

with a specific value of, integrating over, and using the orthogonality property of Legendre polynomials in Eq.(6.43), one gets (6.70) For carrying out the integral, we use Rodrigues' formula in Eq.(6.28), (6.71) where we have integrated by parts. For the remaining integral, we integrate by parts, (6.72) and repeat it number of times to get (6.73) Substituting this in Eq.(6.71), we get (6.74) where the expression for is Rayleigh formula for spherical Bessel function of the first kind. Finally, we have (6.75) An important property is the threshold behaviour. Using the power series expansion for, we get

(6.76) where the leading term is from, and double factorial means alternate terms,. Another important property is the asymptotic behaviour, (6.77) Alternatively, we can approach the problem by considering the radial part in Eq.( 6.8) with, (6.78) Taking, it can be written in the form (6.79) For, the solutions are (6.80) For general, we consider (6.81) Substituting these in Eq.(6.79) leads to (6.82) Taking the derivative with respect to, one gets,

(6.83) Substitute, and divide by to obtain (6.84) as the solution to Eq.(6.82) with replaced by. Therefore starting with and repeating the process in Eq.(6.84) number of times, we get (6.85) where we have used the relations in Eqs.(6.80) and (6.81). Here the solutions with are the same as the spherical Bessel function solutions in Eq.(6.74), and the solutions with are the Neumann functions which are singular at. One can take a combination of these, for example for which the solutions in Eq.(6.85) are the Hankel functions. 6.6 Particle in a spherical box As a first application, we will consider the simple case of a particle with mass in a spherical box, (6.86) The radial solutions inside the box are (6.87) which are non-singular at. Since the potential shoots up to infinity at, the wave function must vanish at, (6.88) For the case of, this implies

(6.89) where the constant is determined from the normalization condition. For the case of, the radial wave function in Eq.(6.87), and the boundary condition lead to (6.90) We can obtain approximate solution to this by considering (6.91) We also consider a particle of mass in a shell potential (6.92) For this case, the radial solution is a combination of the two solutions in Eq.(6.85), (6.93) where and are the Bessel and Neumann functions in Eq.(6.85). For the case of, we can write it as (6.94) The boundary conditions at imply (6.95)

6.7 Positive energy states Positive energy states are of great importance in the description of scattering of a particle or a beam of particles. Here, we must carefully incorporate the appropriate asymptotic behaviour of the wave function. For an incoming beam of particles of mass, with momentum, the wave function at large is of the form (6.96) with ingoing beam flux density. The outgoing scattered particle flux density is (6.97) where area is the scattered part of the wave function. Multiplying the flux density by the in the radial direction, we get the radial, particle flux (6.98) where is the differential cross section which is the scattered particle flux per unit solid angle, per unit incoming flux density. It is convenient to develop the general wave function in terms of partial waves which are eigenstates of angular momentum, and compare with the asymptotic form in Eq.(6.96), (6.99) where the first term is the expansion of in Eq.(6.75) and the second term is the expansion of the scattered wave in Eq.(6.96) in terms of Legendre polynomials. Using the asymptotic behaviour of in Eq.(6.77), we get (6.100) The general approach then is to solve the Schroedinger equation for for appropriate energy and potential, take limit and determine and :

(6.101) where and are the Bessel and Neumann functions in Eq.(6.85). Comparing the two expressions in Eqs.(6.100) and (6.101), for and terms, one gets (6.102) which implies that (6.103) This leads to (6.104) As a simple application we consider scattering by a hard sphere for which (6.105) In this case, the wave function for with is of the form (6.106) where and are the Bessel and Neumann functions in Eq.(6.85). Now the boundary condition at implies that (6.107) For, this gives (6.108) Now the threshold behaviour in Eq.(6.76) implies that the term in Eq.(6.107) dominates for, so that

(6.109) 6.8 Particle in a 3-D S.H.O. potential For a particle of mass in an isotropic S.H.O. potential in 3-D, one has the Hamiltonian (6.110) The separable solutions are related to the solutions in Eq(4.94), Eq.(4.102) for 1-D and in Eq.(5.19) for 2-D, with the energy being the sum of the energies and the wave function being the product of the wave functions in each coordinate, (6.111) where takes the values corresponding to coordinates. Here are the Hermite polynomials in Eq.(4.98), and are the confluent hypergeometric functions in Eq.(4.89). There is considerable amount of degeneracy in these states. For given, we take a fixed value of with taking values with degeneracy, and sum over to get total degeneracy, (6.112) It is interesting and important to consider the solutions in terms of spherical coordinates. For the Schroedinger equation in Eq.(6.6) with, the angular parts are the spherical harmonics in Eq.(6.38), and the radial part satisfies Eq.(6.8), (6.113) (6.114)

Using these in Eq.(6.113) leads to (6.115) Since this equation is invariant under, the solutions can be taken to be odd or even functions of, and we consider solutions of the form (6.116) Substituting this in Eq.(6.115), we get (6.117) Taking the coefficient of leads to (6.118) Taking and noting that, this equation implies (6.119) For solutions non-singular at, one has, and factorising the denominator in Eq.(6.118), we obtain (6.120) Comparing this with the ratio in Eq.(4.90) for the confluent hypergeometric functions in Eq.(4.89), one gets (6.121)

Since for, normalizability of the wave function implies that (6.122) It may be noted that the threshold behaviour (6.123) is determined mainly by the singular term in Eq.(6.113). We also note that for the simple harmonic potential in 3-D, the virial relation in Eq.(4.112) implies (6.124) Also, for the Feynman-Hellmann theorem in Eq.(4.117), taking the parameter and, we get from the energy in Eq.(6.122) (6.125) These relations are the same for the s.h.o in 1-D, 2-D, and 3-D. Creation and annihilation operators A very interesting and useful property of the energy eigenstates of the s.h.o. is that they can be related by simple creation and annihilation operators. We start by considering the Hamiltonian in 1-D, (6.126) This relation can be written in the form

(6.127) A very important property is the commutator, (6.128) Using this in the Hamiltonian, one gets (6.129) These properties are easily generalised to 3-D, (6.130) The description in terms of and can be used to relate and develop different states. To be specific, we consider the normalized eigenstates of in 1-D with eigenvalues, (6.131) Now for the states and, one gets (6.132) where we have used the commutation relations in Eq.(6.129). This implies that is an eigenstate of with eigenvalue and is an eigenstate of with eigenvalue

(6.133) This is the reason why and are described as creation and annihilation operators whose operation increases or decreases the energy by. To determine the constants and, we take the norm of the two sides in Eq.(6.133) and use the relations for in Eq.(6.129), (6.134) For the lowest energy state for which, we have in Eq.(6.133), for which Eq.(6.134) implies (6.135) Starting with and repeating the operation by in Eq.(133) number of times, we get (6.136) One can also obtain the ground state wave function from the condition in Eq.(6.133), (6.137) This leads to (6.138) where is the normalization constant: (6.139) Finally, the wave function for the state is obtained from Eq.(6.136) with obtained from Eq.(6.127),

(6.140) with. It is interesting to note that in addition to the isotropic s.h.o potential, we can introduce an additional term in the potential. Effectively, it is an addition to the angular momentum term in the radial equation in Eq.(6.113), and we can take (6.141) This removes the degeneracy in the eigenstates since the changes in are not in integers. 6.9 Particle in a Coulomb potential We now consider a very important case of a particle in a Coulomb potential. This is the main part of the interaction between charged particles, in the interaction of an electron with nuclei. We will specifically consider the negative energy, bound eigenstates of a particle of mass in a potential (6.142) where and are usually the charges of the electron and the nucleus, is positive, and we are using cgs units. For the separable solutions to the Schroedinger equation in Eq.(6.6), with the angular part in terms of spherical harmonics in Eq.(6.38), one has (6.143) with satisfying the radial equation in Eq.(6.8), (6.144) For developing the appropriate solutions, we first consider the asymptotic behaviour for large, where for the bound states. We then take (6.145)

(6.146) Substituting these in Eq.(6.144) leads to (6.147) Substituting a power series solution, we get (6.148) Taking the coefficient of leads to (6.149) For which implies that (6.150) One takes for solutions which are non-singular at. With this, factorization of the denominator in Eq.(6.149) leads to (6.151) Comparing with the ratio in Eq.(4.90) for the confluent hypergeometric function in Eq.(4.89), we get (6.152) Since for, normalizability of the wave function implies that

(6.153) There are several interesting and important properties of the solutions, which may be noted. Threshold behaviour: The general threshold behaviour is (6.154) which is determined mainly by the singular terms in Eq.(6.144).Degeneracy: For a given principal quantum number, the can take values, and for each there are possible components of the angular momentum. Therefore the degeneracy for a given energy with principal quantum number is (6.155) where we have put in an extra factor of to take into account the two spin components for the spin states of the electron. Scaling property: We start with the Schroedinger equation in Eq.(6.3), multiply it by, and subject it to a scale transformation, leading to (6.156) Taking leads to (6.157) This is equivalent to the original equation with. It therefore implies (6.158) Our results in Eq.(6.153) are consistent with this scaling property. Virial relation: The virial relation in Eq.(4.112) implies for the Coulomb potential in Eq.(6.142),

(6.159) Feynman-Hellmann theorem: For using the Feynman-Hellmann theorem in Eq.(4.117), we take the parameter, and, to get from the energy in Eq.(6.153), (6.160) which are similar to the virial relations in Eq.(6.159). Additional term: As in the case of s.h.o, an additional term is equivalent to changing the coefficient of the angular momentum term in the radial equation in Eq.(6.144). We can take (6.161) This removes the degeneracy in the eigenstates of energy since the changes in are not integers.