Quantum Mechanics Zhiguo Wang Spring, 2014

Approximate Methods in Quantum Mechanics * We have found EXACT solutions for example for the case of the potential well, the OSCILLATOR and …… * In MOST situations however the Schrödinger equation CANNOT be solved exactly and it is necessary instead to resort to NUMERICAL or APPROXIMATE methods * Among the approaches that are available here we will discuss the following  Time-independent and time-dependent PERTURBATION THEORY  The WKB APPROXIMATION for slowly-varying potentials  The VARIATIONAL METHOD for ground-state determination Quantum Mechanics, Tongji University

Chap.9 Quantum Mechanics, Tongji University

TIME-INDEPENDENT PERTURBATION THEORY 6.1 Nondegenerate Perturbation Theory 6.2 Degenerate Perturbation Theory 6.3 The Fine Structure of Hydrogen 6.4 The Zeeman Effect 6.5 Hyperfined Splitting 6.6 Summary Quantum Mechanics, Tongji University

Matrix Representation of Operators solving the Schrödinger equation ( DIFFERENTIAL form) will be more convenient to utilize a MATRIX form * To develop this matrix form we begin by rewriting the SE as * Next we note that the wavefunction can be written as a linear SUPERPOSITION of the eigenfunctions of some arbitrary operator  The summation here is appropriate for a system with discrete eigenfunctions and for an UNCONFINED particle the discrete sum would be replaced by a CONTINUOUS integral Quantum Mechanics, Tongji University

Matrix Representation of Operators Next we introduce Eq. S.2 into Eq. S.1 so that the SE becomes * Next we multiply from the left by m* and integrate to obtain  The integral on the RHS of Eq. S.4 is just the orthonormality condition for the eigenfunctions and immediately reduces to mn  To simplify the form of the LHS of Eq. S.4 we define the so-called MATRIX ELEMENT Hmn Quantum Mechanics, Tongji University

Matrix Representation of Operators Then SE can be rewritten as * This is now an eigenvalue equation in which the Hamiltonian is expressed in MATRIX form  Note that the DIMENSION of the matrix form of H is determined by the number of unique eigenfunctions and can often be INFINITE! Quantum Mechanics, Tongji University

Matrix Representation of Operators To determine the energy eigenvalues from Eq. S.7 we rewrite it as * The condition for this equation to have nontrivial solutions is that its determinant should VANISH * For an n  n matrix solution of this determinant yields n INDEPENDENT simultaneous equations and thus n independent solutions for the energy Quantum Mechanics, Tongji University

Matrix Representation of Operators We know that the operators must be HERMITIAN to ensure that their eigenvalues are real * The corresponding matrix representation of any operator must similarly ALSO be Hermitian * A Hermitian matrix may be DEFINED as one whose elements are related according to The notation < m | H | n > for the matrix elements is due to DIRAC and < m | is referred to as the BRA while | n > is referred to as the KET * Note that the Dirac form keeps only the indices n and m that label the eigenfunctions  Quantum Mechanics, Tongji University

Example Determine the eigenvalues of the NON-HERMITIAN operator A which may be written in matrix form as NOTE THAT THESE EIGENVALUES ARE NOT REAL … THE OPERATOR A IS NOT HERMITIAN! Quantum Mechanics, Tongji University

Matrix Representation of Operators While we have expressed the Hamiltonian in terms of matrix elements involving the eigenfunctions n of an ARBITRARY operator it is natural to use the eigenfunctions of the Hamiltonian itself so that the matrix elements become * This matrix is DIAGONAL since all off-diagonal elements are equal to ZERO * It is possible that OTHER operators may also be diagonal when expressed using the eigenfunctions that diagonalize the Hamiltonian  This can only happen if the two operators COMMUTE. Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory The starting point in our development of perturbation theory is the eigenvalue equation for the EXACT system * While the solutions to this problem are known we wish to find the eigenvalues (En) and eigenfunctions (n) of the perturbed system * The basic idea of perturbation theory is to EXPAND the energy and wavefunctions of the perturbed system in powers of the small potential H’  For this purpose we write H as LATER ON WE WILL SET l = 1 Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory Perturbation theory: A small perturbation modifies the wave functions and energies of the system. Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory Next we write the eigenvalues and eigenfunctions of the PERTURBED system as * The idea is to now calculate the MINIMUM number of terms in this expansion that are necessary to achieve satisfactory approximations for En and n nth EIGENVALUE OF H nth EIGENFUNCTION OF H ZEROTH ORDER TERM SECOND ORDER CORRECTION FIRST ORDER CORRECTION Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory Substituting Eqs. 6.4 & 6.5 into Eq. 6.3 yields * In order for this equation to hold the coefficients of common powers on both sides of the equation MUST be EQUAL to each other  The reason for introducing the constant  now becomes clear since it helps us to identify the common powers in Eq. 6.6 Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory * For the first three powers of  this approach leads to the following relations Eq. 6.7 shows that to ZEROTH order the eigenvalues and eigenfunctions of the perturbed system are just those of the UNPERTURBED system Quantum Mechanics, Tongji University

Time-Independent Perturbation Theory * To obtain the NEXT order approximation we substitute Eqs. 6.10 into Eq. 6.8 * Next we write the 1st-order correction to the nth eigenfunction as the LINEAR SUM Then rewrite this last equation in the following form Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory  The LHS of this equation then VANISHES to yield the important result THE FIRST-ORDER CORRECTION TO THE ENERGY IS JUST THE EXPECTATION VALUE OF THE PERTURBATION … MAKES SENSE! To determine the first-order correction to the wavefunction we multiply Eq. 6.15 by m* (m  n) then integrate to obtain * In this way we can write the first-order correction to the wavefunction as Note here that the coefficient ann(1) only affects the normalization and so can be taken as ZERO Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory * Next we can proceed to obtain the SECOND-ORDER corrections to the eigenvalues and eigenfunctions by introducing our results into Eq. 6.9 Using the results we have developed thus far for zeroth- and first-order terms Eq. 6.9 may be rewritten as Similar to our discussion of the first-order corrections we EXPAND the second-order eigenfunction correction as Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory Introducing this form into Eq. 6.19 then gives To obtain the second-order energy correction we next multiply Eq. 6.21 by n* and then integrate which yields We can now SUMMARIZE our results for the expansion of the energy to second order and the expansion of the wavefunction to first order Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory Some of the IMPLICATIONS of our analysis * Previously we mentioned that perturbation H’ to the exact Hamiltonian should be a SMALL quantity  We now see that this requires the matrix elements H’nk to be much smaller than the energy denominator (H’nk << E0n – E0k)  In other words the COUPLING between the states induced by the perturbation should be much SMALLER than the separation between the original energy levels A problem clearly arises with the forgoing discussion if the energy levels of the unperturbed system are DEGENERATE since the denominator in Energy then VANISHES Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory * The approach we have taken here is referred to as NON-DEGENERATE perturbation theory and a different approach will be developed to deal with degenerate systems * The first-order energy correction H’nn may be of EITHER sign but often VANISHES due to the symmetry of the wavefunctions that appear in its integral  The second term correction then determines the energy and in the case of the GROUND STATE level this correction is always NEGATIVE since E0n < E0k  This is easy to understand since that the perturbation MIXES other states into the ground-state wavefunction and the system can use this freedom to LOWER its energy Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory Example: Charged harmonic oscillator in electric field: Perturbation and exact result. We have already known the zeroth-order states and energy The first order perturbation to the energy is evidently zero Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory As for the first order correction to the wave function, we need calculate the matrix element thus Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory where we have used The second order correction to the energy is which means the perturbation always lower the energy by an amount. On the other hand this example can be solved exactly. The potential energy can be rewritten as which is just a shift of the harmonic potential Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory Example 2 Perturbation in infinite square well. The unperturbed wave functions for the infinite square well are which satisfy We perturb the system in three different ways (Figure) and solve the equation with Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory 1) Suppose first that we perturb the system by simply raising the floor of the well by a constant amount V0. In that case H’ = V0 and the first-order correction to the energy of the n-th state is The corrected energy levels, then, are they are simply lifted by the amount V0. The only surprising thing is that in this case the first-order theory yields the exact answer. For a constant perturbation all the higher corrections vanish. Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory 2) If the perturbation extends only half-way across the well, then In this case every energy level is lifted by V0/2. That's not the exact result, presumably,but it does seem reasonable, as a first order approximation. Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory 3) we put a -function bump in the center of the well For even n the wave function is zero at the location of the perturbation (x = a/2), so it never feels H’. That is why the energies are not perturbed for even n. The correction to the ground state is Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory To find this we need The first 3 nonzero terms will be m = 3; 5; 7. Meanwhile so Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory The second order corrections En(2) can be calculated by explicitly summing the series. We know Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory To sum the series, note that For n = 1 Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory For n = 3 Quantum Mechanics, Tongji University

6.1 Time-Independent Perturbation Theory Problem 6.3 ,6.30 Quantum Mechanics, Tongji University

6.2 Degeneracy perturbation theory Assumed so far that all eigenfunctions are non-degenerate i.e. Q|n> = qn|n> with qn ≠ qm for n ≠ m Proof of orthonormality required qn ≠ qm and <m|n> = mn Degenerate states with qn = qm have <m|n> ≠ 0 for m ≠ n … this leads to problems with perturbation theory! However, if we have d degenerate states which form a complete set it is possible to construct a new complete set of eigenfunctions which are orthogonal … … this procedure is known as Schmidt orthogonalisation Quantum Mechanics, Tongji University

Schmidt orthogonalisation 6.2 Degeneracy perturbation theory Schmidt orthogonalisation Any linear combination of the degenerate states produces the same eigenvalue …… if Q|d> = q|d> then q is the same for all |d> as they are degenerate then if | > =  d cd|d> …… Q | > = Q d cd|d> = d cd Q|d> = d cd q|d> = q | > suppose there are 3 degenerate states |1>, |2> and |3> can construct a state |2’> orthogonal to |1> … |2’> = S12|1> - |2> where S12 = <1|2> proof: <1|2’> = S12<1|1> - <1|2> = S12 - S12 = 0 since <1|1>=1 Quantum Mechanics, Tongji University

6.2 Degeneracy perturbation theory Thus |2’> is orthogonal to |1> , although choice of |2’> is not unique A 3rd function |3’> orthogonal to these two can be similarly shown to exist ... |3’> = S13|1> + S23|2’> - |3> where S13 = <1|3> S23 = <2’|3>/<2’|2’> … so |1> , |2’> and |3’> now form an orthogonal set of degenerate eigenfunctions can extend this method to a system with any number of degenerate eigenfunctions … … all results which depend on orthogonality can now be extended to the degenerate case provided that such an orthogonal set can first be constructed Quantum Mechanics, Tongji University

6.2 Degeneracy perturbation theory 6.2.1 Twofold Degeneracy Suppose that Note that any linear combination of these states, is still an eigenstate of H0, with the same eigenvalue E0 Typically, the perturbation (H') will "break" the degeneracy: As we increase  (from 0 to 1), the common unperturbed energy E0 splits into two. Quantum Mechanics, Tongji University

6.2 Degeneracy perturbation theory When we turn off the perturbation, the "upper" state reduces down to one linear combination of a and b, and the "lower" state reduce, to some other linear combination, but we don't know a priori what these "good" linear combinations will be. For this reason we can't even calculate the first-order energy because we don't know what unperturbed states to use. Quantum Mechanics, Tongji University

6.2 Degeneracy perturbation theory For the moment, therefore, let's just write the "good" unperturbed states in the general form keeping  and  adjustable. We want to solve the Schrodinger equation, H = E, with H = H0 + H' and collecting like powers , as before, we find Quantum Mechanics, Tongji University

6.2 Degeneracy perturbation theory But, so the first terms cancel; at order , we have Taking the inner product with a0: Because H0 is Hermitian, the first term on the left cancels the first term on the right. Considering the orthonormality condition Equation, we obtain Quantum Mechanics, Tongji University

6.2 Degeneracy perturbation theory Similarly, the inner product with b0 yields Then we get If  is nonzero, then This is the fundamental result of degenerate perturbation theory; the two roots correspond to the two perturbed energies. Quantum Mechanics, Tongji University

6.2 Degeneracy perturbation theory Quantum Mechanics, Tongji University

6.2 Degeneracy perturbation theory 6.2.2 Higher-Order Degeneracy the two-fold degeneracy case is easy to generalize. In matrix form: the E1 is nothing but the eigenvalues of the W-matrix; In the case of n-fold degeneracy, we look for the eigenvalues of the nn matrix In the language of linear algebra, finding the "good" unperturbed wave functions amounts to constructing a basis in the degenerate subspace that diagonalizes the perturbation H'. Quantum Mechanics, Tongji University

6.2 Degeneracy perturbation theory Example: Consider the three-dimensional infinite cubical well The stationary states are where nx, ny, and nz are positive integers. The corresponding allowed energies are Notice that the ground state (111) is nondegenerate; its energy is Quantum Mechanics, Tongji University

6.2 Degeneracy perturbation theory But the first excited state is (triply) degenerate: Now let's introduce the perturbation This raises the potential by an amount V0 in one quarter of the box. The first-order correction to the ground state energy is given by Quantum Mechanics, Tongji University

6.2 Degeneracy perturbation theory For the first excited state we need the full machinery of degenerate perturbation theory. The first step is to construct the matrix W. The diagonal elements are the Quantum Mechanics, Tongji University

6.2 Degeneracy perturbation theory But the z integral is zero (as it will be also for Wac), so Finally, The characteristic equation for W (or rather, for 4W/V0, which is easier to work with) is the eigenvalues are Quantum Mechanics, Tongji University

6.2 Degeneracy perturbation theory To first order in . then, where E0 is the (common) unperturbed energy. The perturbation lifts the degeneracy, splitting E0 into three distinct energy levels. Notice that if we had naively applied nondegenerate perturbation theory to this problem, we would have concluded that the first-order correction is the same for all three states, and equal to V0/4--which is actually correct only for the middle state. Meanwhile, the "good" unperturbed states are linear combinations of the form, Quantum Mechanics, Tongji University

6.2 Degeneracy perturbation theory where the coefficients (,,) form the eigenvectors of the matrix W: Thus the "good" states are Quantum Mechanics, Tongji University

6.2 Degeneracy perturbation theory Problem 6.7 , 6.9 Quantum Mechanics, Tongji University

6.3 the fine structure of hydrogen In our study of the hydrogen atom But this is not quite the whole story. To correct for the motion of the nucleus: Just replace m by the reduced mass. More significant is the so-called fine structure, which is actually due to two distinct mechanisms: a relativistic correction, and spin-orbit coupling. fine structure constant Lamb shift, associated with the quantization of the Coulomb field, hyperfine structure due to the magnetic interaction between the dipole moments of the electron and the proton. Quantum Mechanics, Tongji University

6.3 the fine structure of hydrogen Table 6.1: Hierarchy of corrections to the Bohr energies of hydrogen. 6.3.1 The Relativistic Correction kinetic energy: is the classical equation for kinetic energy; the relativistic formula Quantum Mechanics, Tongji University

6.3 the fine structure of hydrogen The lowest-order s relativistic contribution to the Hamiltonian is Quantum Mechanics, Tongji University

6.3 the fine structure of hydrogen In first-order perturbation theory, the correction to En is given by the expectation value of H' in the unperturbed state Since SE, For hydrogen atom En is the Bohr energy of the state in question. Quantum Mechanics, Tongji University

6.3 the fine structure of hydrogen we need the expectation values of 1/r and 1/r2 in the (unperturbed) state nlm eliminating a Notice that the relativistic correction is smaller than E, by a factor of E/mc2 2 10-5. Quantum Mechanics, Tongji University

6.3 the fine structure of hydrogen Noticed: in the calculation used nondegenerate perturbation theory even though the hydrogen atom is highly degenerate. Reason: the perturbation is spherically symmetrical, it commutes with L2 and Lz. Moreover, the eigenfunctions of these operators (taken together) have distinct eigenvalues for the n2 states with a given En. The wave functions nlm are "good" states, so as it happens the use of nondegenerate perturbation theory was legitimate. Quantum Mechanics, Tongji University

6.3 the fine structure of hydrogen 6.3.2 Spin-Orbit Coupling Imagine the electron in orbit around the nucleus; from the electron's point of view, the proton is circling around it. This orbiting positive charge sets up a magnetic field B in the electron frame, which exerts a torque on the spinning electron, tending to align its magnetic moment () along the direction of the field. The Hamiltonian is The Magnetic Field of the Proton. Quantum Mechanics, Tongji University

6.3 the fine structure of hydrogen The Magnetic Dipole Moment of the Electron. If the mass of the ring is m, its angular momentum is the moment of inertia (mv2 times the angular velocity (2/ T): gyromagnetic ratio is independent of r (and T). As long as the mass and the charge are distributed in the same manner (so that the charge-to-mass ratio is uniform), the gyromagnetic ratio will be the same for each ring, and hence also for the object as a whole. Quantum Mechanics, Tongji University

6.3 the fine structure of hydrogen That was a purely classical calculation, however; the electron's magnetic moment is twice the classical answer: The "extra" factor of 2 was explained by Dirac in his relativistic theory of the electron. The Spin-Orbit Interaction. Thomas precession. In the presence of spin-orbit coupling, the Hamiltonian no longer commutes with L and S, so the spin and orbital angular momenta are not separately conserved. However, H’so does commute with L2, S2, and the total angular momentum Quantum Mechanics, Tongji University

6.3 the fine structure of hydrogen therefore the eigenvalues of LS are Meanwhile, the expectation value of 1/r3 Quantum Mechanics, Tongji University

6.3 the fine structure of hydrogen we conclude that It is remarkable, considering the totally different physical mechanisms involved, that the relativistic correction and the spin-orbit coupling are of the same order (En2/mc2), we get the complete fine-structure formula Quantum Mechanics, Tongji University

6.3 the fine structure of hydrogen Fine structure breaks the degeneracy in l (that is, for a given n, the different allowed values of l do not all carry the same energy); the energies are determined by n and j . The azimuthal eigenvalues for orbital and spin angular momentum (ml and ms) are no longer "good" quantum numbers--the stationary states are linear combinations of states with different values of these quantities; the "good" quantum numbers are n, l, s, j, and mj󿼇 Quantum Mechanics, Tongji University

6.3 the fine structure of hydrogen Schematic diagram of 3s and 3p and sodium D lines. Quantum Mechanics, Tongji University

6.3 the fine structure of hydrogen Quantum Mechanics, Tongji University

6.4 THE ZEEMAN EFFECT When an atom is placed in a uniform external magnetic field Bext, the energy level, are shifted. This phenomenon is known as the Zeeman effect. The Zeeman splitting depends critically on the strength of the external field in comparison with the internal field that gives rise to spin-orbit coupling. Quantum Mechanics, Tongji University

6.4 THE ZEEMAN EFFECT if Bext << Bint, then fine structure dominates, and Hz can be treated as a perturbation, if Bext >> Bint, then the Zeeman effect dominates, and fine structure becomes the perturbation. In the intermediate zone, Bext ~ Bint, we need the full machinery of degenerate perturbation theory, and it is necessary to diagonalize the relevant portion of the Hamiltonian "by hand". Quantum Mechanics, Tongji University

6.4.1 Weak-Field Zeeman Effect 6.4 THE ZEEMAN EFFECT 6.4.1 Weak-Field Zeeman Effect the "good" quantum numbers are n, l, j, and mj (in the presence of spin-orbit coupling--L and S are not separately conserved). In first-order perturbation theory, the Zeeman correction is the total angular momentum J = L+S is constant, we do not know the expectation Value of S. since L and S precess rapidly about this fixed vector. the (time) average value of S is just its projection along J: Quantum Mechanics, Tongji University

as choose the z-axis to lie along Bext; then 6.4 THE ZEEMAN EFFECT the Lande g-factor, gj. as choose the z-axis to lie along Bext; then Bohr magneton. Quantum Mechanics, Tongji University

6.4 THE ZEEMAN EFFECT The total energy is the sum of the fine-structure part and the Zeeman contribution. For example. The ground state (n = 1, l = 0, j = 1/2, and therefore gj = 2) splits into two levels: with the plus sign for mj = 1/2, and minus for mj = -1/2. Quantum Mechanics, Tongji University

6.4.2 Strong-Field Zeeman Effect 6.4 THE ZEEMAN EFFECT 6.4.2 Strong-Field Zeeman Effect the "good" quantum numbers are now n, l, ml, and ms (but not j and mj because---in the presence of the external torque--the total angular momentum is not conserved, whereas Lz and Sz are). the "unperturbed" energies are In first-order perturbation theory, the fine-structure correction to these levels is Quantum Mechanics, Tongji University

The relativistic contribution is the same as before. 6.4 THE ZEEMAN EFFECT The relativistic contribution is the same as before. we conclude that The total energy is the sum of the Zeeman part and the fine-structure contribution Quantum Mechanics, Tongji University

6.4.3 Intermediate-Field Zeeman Effect 6.4 THE ZEEMAN EFFECT 6.4.3 Intermediate-Field Zeeman Effect in the case n = 2 and choose as the basis for degenerate perturbation theory the states characterized by l, j, and mj . Using the Clebsch-Gordan coefficients (express |j mj> as a linear combination of |I ml>Is ms>), we have Quantum Mechanics, Tongji University

6.4 THE ZEEMAN EFFECT Quantum Mechanics, Tongji University

In this basis, the complete matrix-W is 6.4 THE ZEEMAN EFFECT In this basis, the complete matrix-W is The first four eigenvalues are displayed along the diagonal; it remains only to find the eigenvalues of the two 2 x 2 blocks. The eigenvalues of the second block are the same, but with the sign of  reversed.The eight energies are listed in Table Quantum Mechanics, Tongji University

6.4 THE ZEEMAN EFFECT Energy levels for the n = 2 states of hydrogen, with fine structure and Zeeman splitting. Quantum Mechanics, Tongji University

6.4 THE ZEEMAN EFFECT Zeeman splitting of the n = 2 states of hydrogen in the weak, intermediate, and strong field regimes. Quantum Mechanics, Tongji University

Normal Zeeman effect of sodium D lines. 6.4 THE ZEEMAN EFFECT Normal Zeeman effect of sodium D lines. Quantum Mechanics, Tongji University

Anomalous Zeeman effect of sodium D lines. 6.4 THE ZEEMAN EFFECT Anomalous Zeeman effect of sodium D lines. Quantum Mechanics, Tongji University

6.5 HYPERFINE SPLITTING The proton itself constitutes a magnetic dipole, though its dipole moment is much smaller than the electron's because of the mass in the denominator According to classical electrodynamics, a dipole sets up a magnetic field So the Hamiltonian of the electron, in the magnetic field due to the proton's magnetic dipole moment, is Quantum Mechanics, Tongji University

6.5 HYPERFINE SPLITTING According to perturbation theory, the first-order correction to the energy is the expectation value of the perturbing Hamiltonian: In the ground state (or any other state for which l = 0) the wave function is spherical symmetrical, and the first expectation value vanishes. Meanwhile. in the ground state. This is called spin-spin coupling because it involves the dot product of two spins. Quantum Mechanics, Tongji University

6.5 HYPERFINE SPLITTING In the presence of spin-spin coupling, the individual spin angular momenta are no longer conserved; the "good" states are eigenvectors of the total spin, the electron and proton both have spin 1/2, In the triplet state (spins "parallel") the total spin is 1, and hence S2 = 2h2; in the singlet state the total spin is 0, and S2 = 0. Thus Quantum Mechanics, Tongji University

Hyperfine splitting in the ground state of hydrogen. Spin-spin coupling breaks the spin degeneracy of the ground state, lifting the triplet configuration and depressing the singlet. The energy gap is The frequency of the photon emitted in a transition from the triplet to the singlet state is Hyperfine splitting in the ground state of hydrogen. Quantum Mechanics, Tongji University

6.5 HYPERFINE SPLITTING the corresponding wavelength is c/v = 21 cm, which falls in the microwave region. This famous "21-centimeter line" is among the most pervasive and ubiquitous forms of radiation in the universe. Quantum Mechanics, Tongji University

End Quantum Mechanics, Tongji University

Summary Quantum Mechanics, Tongji University 86

Summary That’s all. Quantum Mechanics, Tongji University 87

2 The Zeeman Effect Suppose an atom has eigenstates ψnlml with energy levels Enl in the absence of a magnetic field. We have Since ψnlml is an eigenstate of Lz with eigenvalue mlh, Quantum Mechanics, Tongji University

2 The Zeeman Effect Quantum Mechanics, Tongji University

2 The Zeeman Effect Quantum Mechanics, Tongji University

2 The Zeeman Effect The separation between the new levels in the multiplet is given by μBB. For a magnetic field B of a few teslas, the separation is of the order of 10−4 − 10−5 eV which is rather small. Nevertheless, this small splitting results in an observable splitting of the spectral lines of the light emitted or absorbed by the atom. The amount of splitting increases linearly with the strength of the applied field. This effect of spectral-line splitting by a magnetic field is known as the normal Zeeman effect, and was first observed by Zeeman in 1896. (Zeeman shared the 1902 Nobel prize with his teacher Lorentz who suggested him to investigate the effect of magnetic field on atomic spectra.) Quantum Mechanics, Tongji University

2 The Zeeman Effect This normal Zeeman effect is, however, frequently not observed. More often, additional splitting are found or splitting with uneven spacing are seen. This is known as the anomalous Zeeman effect. In the following, we shall see that the anomalous Zeeman effect has its roots in the existence of electron spin. Quantum Mechanics, Tongji University

4 The anomalous Zeeman effect The total magnetic moment of an electron is the sum of its orbital and spin magnetic moments: In Section 2, we discussed the splitting of spectral lines due to μorb, which is known as the normal Zeeman effect. The presence of μspin causes additional splitting of the spectral lines, in addition to those considered in the normal Zeeman effect, and this is the reason for the anomalous Zeeman effect. Quantum Mechanics, Tongji University

4 The anomalous Zeeman effect To illustrate this, we first consider the simple case of hydrogen atoms in an s state with l = 0 and energy En. – If the electron has no spin, then with l = 0, the atoms would be unaffected by a magnetic field. – But as the electron in fact has spin, so even though l = 0, there is a magnetic moment – When a magnetic field is switched on, there is an additional magnetic potential energy – Taking the direction of B to be the z-axis, we have Quantum Mechanics, Tongji University

4 The anomalous Zeeman effect --this implies that the original energy level En becomes so the original energy level is split into two levels with energies – The separation between the two levels is separation of levels = 2μBB (S7) Notice that this separation is twice the value obtained in the normal Zeeman effect. Quantum Mechanics, Tongji University

4 The anomalous Zeeman effect Hence any l = 0 level of the hydrogen atom should be split into two neighbouring levels by a magnetic field. Such splittings were indeed observed, providing strong evidence that the electron has spin with s = 1/2. Moreover, the observed level separation agreed with (S7), confirming the expression for μspin, that is, Quantum Mechanics, Tongji University

4 The anomalous Zeeman effect For the general case of l ≠ 0, the additional magnetic potential energy when the hydrogen atoms are placed in a magnetic field B is: Quantum Mechanics, Tongji University

4 The anomalous Zeeman effect The original energy level En would thus change to This splitting of the energy levels will result in the splitting of spectral lines. The following figure shows an example of the predicted spectrum for the hydrogen atoms initially in n = 2 state. Note that some of the predicted spectral lines are not observed because the corresponding transitions are forbidden by selection rules. Moreover, the spin magnetic moment and the orbital magnetic moment interact with each other. This spin-orbit interaction (will be discussed in the next section) would lead to more complicated spectra. Quantum Mechanics, Tongji University

4 The anomalous Zeeman effect Quantum Mechanics, Tongji University

4 The anomalous Zeeman effect Quantum Mechanics, Tongji University

5 Spin-Orbit Interaction Even in the absence of a magnetic field, many spectral lines are observed to be actually two separate lines that are very close together. This is known as the fine structure. As an example, the first line of the Balmer series of hydrogen is predicted to have wavelength 6563 Å but in reality it consists of two lines around that wavelength but are 1.4 Å apart. Quantum Mechanics, Tongji University

5 Spin-Orbit Interaction This suggests that even in the absence of a magnetic field, the energy levels are actually split with a small spacing of the order 10−4− 10−5 eV. It turns out that the fine-structure doubling is caused by the mutual interaction of the spin and orbital magnetic moments. This interaction is called spin-orbit interaction or spin-orbit coupling. This spin-orbit interaction can be understood in terms of the following classical model. – An electron revolving around a nucleus “sees” the nucleus circling around it, and thus “feels” a magnetic field generated by the circling motion of the nucleus. Quantum Mechanics, Tongji University

5 Spin-Orbit Interaction – This internal magnetic field B is proportional to the orbital frequency of the nucleus as seen by the electron, which in turn is proportional to the orbital angular momentum L of the electron. – As a result, the electron gets an additional magnetic potential energy Due to orbital L Because it is proportional to S ·L, such an interaction is called the spin-orbit interaction. Quantum Mechanics, Tongji University

5 Spin-Orbit Interaction For a state with l ≠ 0, this spin-orbit interaction causes a splitting of the energy level into two levels corresponding to ms = ±1/2, leading to the fine-structure doubling of many spectral lines. As an example, let us consider transitions in which a hydrogen atom in one of its 2p states drops to the ground state. Quantum Mechanics, Tongji University

5 Spin-Orbit Interaction – The transitions from 2p states to the ground state thus produce a doublet of spectral lines. – The separation between the lines is separation of 2p states = 2μBB where B is the magnetic field of the orbiting proton, as seen by the 2p electrons. – One can estimate B classically and gets a value less than but about 1 T (exercise). This gives the separation of the order of 10−4 − 10−5 eV as observed. We note that spin-orbit interaction energy is of the order of the relativistic corrections that we have ignored so far. So a correct analysis of the fine structure of hydrogen needs to be fully quantum mechanical and takes into account of relativity. Nevertheless, our general conclusions are qualitatively correct. Quantum Mechanics, Tongji University

6 Summary You should know: – orbital magnetic moment of the electron – Bohr magneton – spin angular momentum (spin quantum number s and spin magnetic quantum number ms) – spin quantum number s = 1/2 for the electron – spin magnetic moment of the electron – gyromagnetic ratio – the normal and anomalous Zeeman effect – spin-orbit interaction and fine structure doubling Quantum Mechanics, Tongji University