ŠĻą”±į>ž’ łūž’’’÷ųKéc÷vꀀŚ€€Ł€’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’ģ„Įq` ų涱bjbjqPqP mÜ::©&’’’’’’¤“““““““$Ų\\\P¬Œ8…Ų#å¶`†ŗ(Æ"<Æ<Æ<Æ<ÆŖę°|b±@¢ä¤ä¤ä¤ä¤ä¤ä¤ä$ŁåhAč^Čä“ę³<Æ<Æę³ę³Čä““<Æ<ÆŻä6Ż6Ż6Żę³–“<Æ“<Æ¢ä6Żę³¢ä6Ż6ŻV¶ć@““zä<ÆT† 0ÆĶižgĒ\|øD öć ¢äóä0#åäxŸčĄĆŸčzäŸč“zä(¢±v²T6Żl²D°²6¢±¢±¢±ČäČäŽÜX¢±¢±¢±#åę³ę³ę³ę³ŲŲŲ„y\ŲŲŲ\ŲŲŲ““““““’’’’  Physics 406 Introduction to Quantum Mechanics J Kiefer January 2007 © 2007 Table of Contents  TOC \o "1-2" \h \z \u  HYPERLINK \l "_Toc161820693" Table of Contents  PAGEREF _Toc161820693 \h 1  HYPERLINK \l "_Toc161820694" I. Wave Mechanics  PAGEREF _Toc161820694 \h 2  HYPERLINK \l "_Toc161820695" A. Wave Function  PAGEREF _Toc161820695 \h 2  HYPERLINK \l "_Toc161820696" B. The Schrödinger Equation  PAGEREF _Toc161820696 \h 5  HYPERLINK \l "_Toc161820697" C. Wells and Barriers—Non-Free Particles  PAGEREF _Toc161820697 \h 10  HYPERLINK \l "_Toc161820698" D. Harmonic Oscillator  PAGEREF _Toc161820698 \h 17  HYPERLINK \l "_Toc161820699" II. Hydrogen Atom  PAGEREF _Toc161820699 \h 23  HYPERLINK \l "_Toc161820700" A. Schrödinger Equation in Three Dimensions  PAGEREF _Toc161820700 \h 23  HYPERLINK \l "_Toc161820701" B. Angular Momentum  PAGEREF _Toc161820701 \h 25  HYPERLINK \l "_Toc161820702" C. Radial Equation  PAGEREF _Toc161820702 \h 30  HYPERLINK \l "_Toc161820703" D. Spin  PAGEREF _Toc161820703 \h 35  HYPERLINK \l "_Toc161820704" III. Selected Quantum Mechanical Issues  PAGEREF _Toc161820704 \h 39  HYPERLINK \l "_Toc161820705" A. Vector Spaces  PAGEREF _Toc161820705 \h 39  HYPERLINK \l "_Toc161820706" B. Formal Quantum Mechanics  PAGEREF _Toc161820706 \h 43  HYPERLINK \l "_Toc161820707" C. Time Independent Perturbation Theory  PAGEREF _Toc161820707 \h 45  I. Wave Mechanics A. Wave Function We propose to describe the motion or state of a particle by a wave function—a solution to a wave equation. 1. Statistical Interpretation of the Wave Function The wave function is a complex function of position (x) and of time (t), denoted  EMBED Equation.3 . We intend that it contain a complete description of the behavior of a particle, such as an electron. a. Probability density  EMBED Equation.3  The quantity  EMBED Equation.3  is the probability that the particle is to be found in the interval (x,x+dx) at time t. b. Normalization The probability of finding the particle someplace must be 1.0, therefore,  EMBED Equation.3  if the wave function is to represent a physically realistic particle. That is, a wave function must be normalizable. This boils down to the requirement that  EMBED Equation.3  as  EMBED Equation.3 . Some wave functions satisfy the wave equation but are not normalizable. c. Expectation values Imagine a large number of identical but independent regions in space. In each region is a particle described by the wave function,  EMBED Equation.3 . The particles and wave functions are identical. In each of these identical systems we measure the position of the particle, x. The average of all the independent measurements of x is  EMBED Equation.3 . This is called the expectation value of x. Note that this is not the most probable value of x. That occurs where  EMBED Equation.3  is a maximum. The variance of the measurements of x is  EMBED Equation.3 . 2. Momentum The other dynamical variable of interest is the particle momentum,  EMBED Equation.3 . We will want the expectation value of p, namely

. a. Velocity  EMBED Equation.3   EMBED Equation.3  Note that  EMBED Equation.3 . Now, the wave equation is  EMBED Equation.3 . Substitute this for the time derivatives in the velocity. . .  EMBED Equation.3 . Note that  EMBED Equation.3 and that  EMBED Equation.3  as  EMBED Equation.3 . Next, integrate by parts the first term on the right hand side (r.h.s.).  EMBED Equation.3  Integrate by parts again.  EMBED Equation.3  So far, we have  EMBED Equation.3 . However,  EMBED Equation.3  and  EMBED Equation.3  What’s left is the expectation value of the particle velocity  EMBED Equation.3 . Note for the future:  EMBED Equation.3 . b. Momentum operator We define the particle momentum to be  EMBED Equation.3 . We define the momentum operator to be  EMBED Equation.3 . c. Uncertainty principle The principle will be proven rigorously later; for now, we recognize that  EMBED Equation.3 . Therefore, a spread in  EMBED Equation.3  is inversely related to a spread in p. In terms of a localized waveform,   We identify  EMBED Equation.3  with  EMBED Equation.3  and obtain  EMBED Equation.3 . Roughly stated, this is the mathematical origin of the uncertainty principle. The particle position and momentum cannot be “known” simultaneously to arbitrary precision. B. The Schrödinger Equation 1. Wave Equation The one-dimensional wave equation is  EMBED Equation.3  a. Solution The wave equation has solutions of the form  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 . These are all traveling harmonic waves, where the wave number is  EMBED Equation.3  and the angular frequency is  EMBED Equation.3 . (f is the frequency in Hz.) We’ll concentrate on the complex exponential form:  EMBED Equation.3 . Then the derivatives are  EMBED Equation.3  and  EMBED Equation.3 . Evidently,  EMBED Equation.3  Whence we can identify  EMBED Equation.3 . b. Energy and momentum For a free particle,  EMBED Equation.3 , just the kinetic energy. We can express this in terms of the frequency and wave number, since  EMBED Equation.3  and  EMBED Equation.3 :  EMBED Equation.3 . Solving for the frequency, we obtain what is called a dispersion relation.  EMBED Equation.3  If we substitute this for  EMBED Equation.3  in the wave equation, we obtain the following:  EMBED Equation.3  If we multiply and divide the r.h.s. by  EMBED Equation.3 , we can see that the r.h.s. is just the momentum operator squared, divided by 2m.  EMBED Equation.3   EMBED Equation.3  The left hand side (l.h.s.) must be the total energy operator,  EMBED Equation.3 , since for a free particle, the total energy is the kinetic energy. c. Conservative forces For a conservative force acting on a particle,  EMBED Equation.3 , where V(x,t) is the potential energy function. We’ll just add it to the kinetic energy operator on the r.h.s. of the wave equation.  EMBED Equation.3   EMBED Equation.3  This equation is known as the Time Dependent Schrödinger Equation. 2. Time Independent Schrödinger Equation. Let us say that V is independent of time. The potential energy function is constant. a. Separation of variables Assume that  EMBED Equation.3 . Substitute into the Schrödinger equation. The partial derivatives become  EMBED Equation.3  and  EMBED Equation.3 .  EMBED Equation.3  Divide both sides by  EMBED Equation.3 .  EMBED Equation.3  The two sides must equal the same constant, namely the total energy, E.  EMBED Equation.3  and  EMBED Equation.3 . b. Time solution  EMBED Equation.3   EMBED Equation.3  This is an oscillatory solution, as we always get with a wave equation. c. Stationary states  EMBED Equation.3  Consider the probability density.  EMBED Equation.3 . This is constant in time, therefore it is called a stationary state. A stationary state has a definite total energy; the uncertainty  EMBED Equation.3 . Define the Hamiltonian operator  EMBED Equation.3 . The expectation value of this operator is  EMBED Equation.3 a constant. If the wave function,  EMBED Equation.3 , is normalized, then  EMBED Equation.3 . In compact form, the (Time Independent) Schrödinger equation looks like  EMBED Equation.3 . 3. Free Particles and Wave Packets For a free particle, V(x) = 0. The wave equation is just  EMBED Equation.3 . a. General solution  EMBED Equation.3  (time independent)  EMBED Equation.3  (time dependent). These represent harmonic waves, one traveling to the right, the other to the left along the x-axis. The wave number is  EMBED Equation.3  and the angular frequency is  EMBED Equation.3 . Notice that  EMBED Equation.3 ! The free particle wave function is not normalizable. However, this does not mean that a free particle cannot exist. It does mean that a free particle does not have a definite energy, nor can it be considered to have a localizable position. b. Wave packets A localized wave form can be constructed by a superposition of harmonic waves.  EMBED Equation.3  It might look like this:  The wave packet travels with a speed called the group velocity, vg. The individual ripples within the packet travel with a speed called the phase velocity, vp. The phase velocity is the familiar wave speed.  EMBED Equation.3  c. Group velocity The angular frequency of a harmonic wave is  EMBED Equation.3 . This is called a dispersion relation, because it implies that waves of different frequency travel with different wave speeds. Let’s say that the wave packet is composed of harmonic waves with a narrow range of k-values, centered on ko. We might expand  EMBED Equation.3  in a Taylor’s Series about ko, and keep just the first two terms.  EMBED Equation.3  Put this in the  EMBED Equation.3  for  EMBED Equation.3 .  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Evidently, the wave packet slides along the x-axis at the speed  EMBED Equation.3 .  EMBED Equation.3  Notes: i)  EMBED Equation.3  ii) the classical speed of a particle with momentum p is  EMBED Equation.3 , where  EMBED Equation.3  and  EMBED Equation.3  is the deBroglie wavelength. d. Uncertainty relation We have the wave packet and its Fourier transform: Wave packet:  EMBED Equation.3  Transform:  EMBED Equation.3   The details depend on the exact shape of  EMBED Equation.3 . For the sake of simplicity, let’s say that  EMBED Equation.3  for  EMBED Equation.3  and zero elsewhere. Then the transform is  EMBED Equation.3   Take  EMBED Equation.3  to be the width of the central peak, where  EMBED Equation.3 . We find that  EMBED Equation.3 , since  EMBED Equation.3 . For more realistically-shaped wave packets we find that  EMBED Equation.3 . C. Wells and Barriers—Non-Free Particles 1. Square Wells a. Infinite  EMBED Equation.3   We have three (3) distinct regions, because the potential energy function (potential for short) changes discontinuously. So, we have to solve the Schrödinger Equation three (3) times. Fortunately, for  EMBED Equation.3  and  EMBED Equation.3 , the solutions are trivial:  EMBED Equation.3 , since  EMBED Equation.3 . Within the well,  EMBED Equation.3 , so the particle is “free.”  EMBED Equation.3  The general solution is  EMBED Equation.3 . The parameters A, B, and k are determined by the boundary conditions on the  EMBED Equation.3  and by the normalization requirement. That is, we expect that  EMBED Equation.3 . At x = 0:  EMBED Equation.3  At x = a:  EMBED Equation.3 , since we cannot have A and B both be zero. It follows that  EMBED Equation.3  From this we obtain the discrete allowed energy levels for the particle confined in the well.  EMBED Equation.3  Putting in the allowed values of k yields  EMBED Equation.3 , n = 1,2,3,… . The n is known as the principle quantum number. It labels the energy levels, or energy states of the particle. Finally we need to normalize the wave function.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Our solution is  EMBED Equation.3  for the particle inside the well. Properties of the solution: i) even or odd ii) orthonormal iii) form a complete set. Note: While  EMBED Equation.3  is unchanging with time, the  EMBED Equation.3  is not.  EMBED Equation.3  This form is very similar to the case of a vibrating string, fastened at both ends. b. Finite square well  EMBED Equation.3  We find it convenient, but not necessary, to place the origin in the center of the well.  Again, we solve the Schrödinger equation in three regions:  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 . We’ll solve for the so-called bound states whose energies E < 0.  EMBED Equation.3  The solutions will have the same form as for the infinite well, but the k and the coefficients will differ in the three regions.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 , where  EMBED Equation.3  and  EMBED Equation.3 . [We’ll be setting, as in the text,  EMBED Equation.3 .] The boundary conditions are that  EMBED Equation.3  and  EMBED Equation.3  be continuous at x = - a and x = a.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Before proceeding blindly to solve these simultaneous equations for A, B, C, D, E, & F (we only have 4 equations), let us notice that since E < 0, the term  EMBED Equation.3  diverges for x < -a. Therefore the A = 0. Likewise, G = 0. So we really have only four unknowns remaining.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Put ‘em all in the same order. . .  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  This system of simultaneous equations will have a solution if the determinant of the coefficients vanishes.  EMBED Equation.3  Firstly, divide the 1st and 4th columns by  EMBED Equation.3 .  EMBED Equation.3  Secondly, expand on the 1st column.  EMBED Equation.3  Expand the two determinants on their 3rd columns.  EMBED Equation.3  Expand the 2x2 determinants.  EMBED Equation.3  Rearrange.  EMBED Equation.3  Now,  EMBED Equation.3 . Furthermore,  EMBED Equation.3  and  EMBED Equation.3 . So, we can write  EMBED Equation.3   EMBED Equation.3  Divide by  EMBED Equation.3 .  EMBED Equation.3  The left hand side is a quadratic in  EMBED Equation.3 .  EMBED Equation.3  Use the quadratic formula:  EMBED Equation.3   EMBED Equation.3  We have, after all that, two sets of solutions. We solve for k graphically.   An allowed energy level occurs where these curves intersect. Properties of the solution: i) always at least one solution ii) the allowed energies are discrete iii) the number of solutions is finite, depending on a. iv) +/- solutions alternate in energy (odd/even, as with the infinite well). 2. Potential Step  a. Solution of the time-independent equation  EMBED Equation.3  In this instance, we have two regions: x < 0 and x > 0. For x < 0, V = 0, so at once we can write  EMBED Equation.3 , as usual. For x > 0, V = Vo. We will assume the same mathematical form for the solution, but we may or may not get oscillatory behavior.  EMBED Equation.3  Before applying the boundary conditions, we’ll solve for the  EMBED Equation.3 .  EMBED Equation.3   EMBED Equation.3  Evidently,  EMBED Equation.3 . b. Boundary conditions The boundary conditions are that  EMBED Equation.3  and that  EMBED Equation.3 . We obtain two equations and four unknowns.  EMBED Equation.3  However, we have additional information. In the region x < 0, we have traveling harmonic waves, one incident from the left, the other reflected from the step and traveling toward the left. These cannot be normalized, so we’ll just set A = 1. On the other hand, in the region x > 0, the D = 0, because there is no additional barrier or interface to cause another reflection.  EMBED Equation.3  Firstly, divide them.  EMBED Equation.3  Substitute this result back into the first equation.  EMBED Equation.3  c. Cases Now, consider the cases E > Vo and E < Vo. i) E > Vo:  EMBED Equation.3  is a traveling wave, but of differing wave number from the incident wave. ii) E < Vo:  EMBED Equation.3  is imaginary. Therefore  EMBED Equation.3  is a decaying exponential function.  EMBED Equation.3  Here’s the funny thing. The particle is not moving to the right for x > 0, but yet  EMBED Equation.3 . There is some non-zero probability that the particle will be observed at an x > 0. Classically, this is impossible. 3. Barrier Tunneling  In this case, we have three regions. We can take advantage of our findings for the potential step.  EMBED Equation.3  As before,  EMBED Equation.3  and  EMBED Equation.3 . We apply the boundary conditions at x = o and at x = a.  EMBED Equation.3  Solve these for A, B, R, and T any way you can. . . Qualitatively, we have the same two cases: E > Vo and E < Vo.   One thing we see is that even if E < Vo, the particle probability density is not zero on the far side of the barrier. D. Harmonic Oscillator 1. Schrödinger Equation a. Potential energy  EMBED Equation.3   b. Hamiltonian  EMBED Equation.3  Schrödinger equation  EMBED Equation.3  The total energy operator is called the Hamiltonian:  EMBED Equation.3 . There are two ways to solve this equation. 2. First Way—the Algebraic Method a. Ladder operators Recall that  EMBED Equation.3 . Thus  EMBED Equation.3  and EMBED Equation.3 . We rewrite the Schrödinger equation as  EMBED Equation.3  Factor the thing in the square bracket.  EMBED Equation.3  Note that the order is important. If we write out the product, we must write it thusly:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Evidently, the operators don’t factor quite as numbers do-- EMBED Equation.3 . Let  EMBED Equation.3  and  EMBED Equation.3 . Evidently,  EMBED Equation.3  The Schrödinger equation becomes  EMBED Equation.3 , which looks neater, anyway. b. Raising and lowering  EMBED Equation.3  Suppose that  EMBED Equation.3 , then  EMBED Equation.3  That is, if  EMBED Equation.3  is a solution to the Schrödinger equation with energy E, then  EMBED Equation.3  is also a solution, with energy  EMBED Equation.3 . Similarly,  EMBED Equation.3  produces the solution with energy  EMBED Equation.3 . Hence the terms raising and lowering operators. c. Ground state The state having the lowest energy is called the ground state. We label it  EMBED Equation.3 . Since it is the state of lowest energy, evidently  EMBED Equation.3 .  EMBED Equation.3   EMBED Equation.3  The ground state energy is obtained from the Schrödinger equation.  EMBED Equation.3  Notice that not only is the energy discrete, but the ground state energy is not zero, but  EMBED Equation.3 . d. Energy levels  EMBED Equation.3  The energy levels are uniformly spaced, and n is the principle quantum number. The energy state functions are obtained from the raising operator:  EMBED Equation.3 . 3. Second Way—the Analytical Method a. Asymptotic behavior of  EMBED Equation.3   EMBED Equation.3  If  EMBED Equation.3 , then  EMBED Equation.3 , which has solutions of the form  EMBED Equation.3 .  EMBED Equation.3  because that term diverges as  EMBED Equation.3 . So, let us propose that  EMBED Equation.3 . b. Sturm-Liouville equation The Sturm-Liouville equation is one of those “well known” differential equations for which people worked out the solutions before television was invented. By substituting the assumed  EMBED Equation.3  into the Schrödinger equation, and making a change of variable, we’ll produce a differential equation for that unspecified function,  EMBED Equation.3 . Let  EMBED Equation.3 . Then  EMBED Equation.3 .  EMBED Equation.3  The first and second derivatives of  EMBED Equation.3  with respect to y are  EMBED Equation.3  The  EMBED Equation.3  divides out, so we have an equation for the h(y).  EMBED Equation.3  Where we have set  EMBED Equation.3 . c. Solution of the Sturm-Liouville equation The standard tack is to assume a series solution:  EMBED Equation.3 . Plug this into the S-L equation, and we obtain a recursion relation for the coefficients. We’d like to have all the sums start at the same place, as well.  EMBED Equation.3   EMBED Equation.3  Collect the coefficients of  EMBED Equation.3  and set equal to zero.  EMBED Equation.3  Solve for  EMBED Equation.3 . This result is a recursion relation for two sequences of coefficients. Namely  EMBED Equation.3  and  EMBED Equation.3 . However, an infinite series is not normalizable. So the series must terminate at some finite number of terms; say at  EMBED Equation.3 . That is  EMBED Equation.3  and all succeeding  EMBED Equation.3  also. The h(y) is a finite polynomial of degree n. Of course, n is going to be the principle quantum number. Through the  EMBED Equation.3 , the n is related to the total energy, E. Set  EMBED Equation.3 , solve for E when j = n.  EMBED Equation.3  These are the energy levels for the harmonic oscillator. We have to solve finally for the h(y). d. Hermite polynomials Of course, we have to get ao and a1 in order to start the recursions for the rest of the coefficients. For that we use the normalization condition. After that, with  EMBED Equation.3 ,  EMBED Equation.3 . The results are the coefficients of the Hermite Polynomials, Hn. The first few Hermite polynomials are shown in Table 2.1 in the text. These form a complete orthogonal set of functions. This means that an arbitrary function can be expanded as a series of Hermite polynomials, just as any vector can be expanded in terms of the Cartesian unit vectors. However, there are an infinite number of Hermite polynomials; they may be regarded mathematically as unit vectors in some infinite-dimensional space. e. Harmonic oscillator wave functions Finally, we put the pieces together to form the harmonic oscillator wave functions.  EMBED Equation.3  See Fig. 2.5 in the text. II. Hydrogen Atom A. Schrödinger Equation in Three Dimensions 1. Hamiltonian Operator  EMBED Equation.3  a. Momentum operators  EMBED Equation.3  b. Commutation relations A commutator is a particular combination of two operators:  EMBED Equation.3 . In general,  EMBED Equation.3 , that is  EMBED Equation.3 . If  EMBED Equation.3 , then the two operators are said to commute. Take a look at  EMBED Equation.3  We say that  EMBED Equation.3 . Now, try  EMBED Equation.3  Repeat for all possible combinations of  EMBED Equation.3 . c. Total energy Classically,  EMBED Equation.3 . The corresponding operator is  EMBED Equation.3 . This is referred to as the Hamiltonian operator. 2. Spherically Symmetric Potential Energy Function(s)  EMBED Equation.3 , with no dependence on  EMBED Equation.3  or  EMBED Equation.3 . a. Spherical polar coordinates  EMBED Equation.3  b. Separate the variables  EMBED Equation.3  Assume that  EMBED Equation.3  and substitute that into the Schrödinger equation.  EMBED Equation.3  Divide both sides by RY and multiply both sides by  EMBED Equation.3 , and collect terms.  EMBED Equation.3  The quantities in the  EMBED Equation.3  must separately be equal to a constant,  EMBED Equation.3 , else the equation won’t be valid for all  EMBED Equation.3 . Therefore, we have two equations, Angular equation:  EMBED Equation.3  Radial equation:  EMBED Equation.3  These have to be solved separately. B. Angular Momentum Classically,  EMBED Equation.3 .  1. Angular Momentum Operators a. Cartesian components of  EMBED Equation.3   EMBED Equation.3  Writing the momentum components as operators, we obtain the angular momentum operators:  EMBED Equation.3  b. Commutation relations  EMBED Equation.3  You can verify these by writing them out. The other angular momentum operator we will need is the total angular momentum squared, L2.  EMBED Equation.3  2. Angular Momentum Ladder Operators Let us assume that f is an eigen function of both the  EMBED Equation.3  and  EMBED Equation.3  operators. That is,  EMBED Equation.3 , .where  EMBED Equation.3  are numbers, known as the eigen values. a. Raising & lowering Define the angular momentum ladder operators as  EMBED Equation.3 . Claim: If  EMBED Equation.3 , then  EMBED Equation.3  also. Proof: Consider the following—  EMBED Equation.3  Now, consider  EMBED Equation.3  Similarly,  EMBED Equation.3 . It follows that  EMBED Equation.3 , as well. Therefore,  EMBED Equation.3 . Tah dah. Or q.e.d. At the same time,  EMBED Equation.3  is an eigenfunction of  EMBED Equation.3  with eigenvalue  EMBED Equation.3  .  EMBED Equation.3  The remaining task is to evaluate the  EMBED Equation.3  and the  EMBED Equation.3  . b. Angular momentum quantum numbers As a vector component,  EMBED Equation.3  cannot exceed  EMBED Equation.3  . Therefore, there are both upper and lower limits on  EMBED Equation.3  . Let’s call the maximum, or top  EMBED Equation.3 . Then,  EMBED Equation.3  What is  EMBED Equation.3  in terms of  EMBED Equation.3 ? It’s quickest if we wrote  EMBED Equation.3  in terms of the  EMBED Equation.3 , because we “know”  EMBED Equation.3  and  EMBED Equation.3  and  EMBED Equation.3  , but not  EMBED Equation.3  , etc.  EMBED Equation.3  Solve for  EMBED Equation.3  and plug in  EMBED Equation.3 .  EMBED Equation.3  Thus, we can identify  EMBED Equation.3 . By a similar token, there is a minimum value of  EMBED Equation.3 ; call it  EMBED Equation.3  for the time being.  EMBED Equation.3  Evidently,  EMBED Equation.3  Therefore, we must have  EMBED Equation.3 . In conclusion, we have the eigenvalues of  EMBED Equation.3  differing by  EMBED Equation.3  and running from  EMBED Equation.3  to  EMBED Equation.3 . The eigenvalues of  EMBED Equation.3  are equal to  EMBED Equation.3 . We’ll label the eigenstates according to the quantum numbers  EMBED Equation.3  and  EMBED Equation.3 :  EMBED Equation.3  The azimuthal quantum number is  EMBED Equation.3 ; the orbital quantum number is  EMBED Equation.3 . The next task is to solve the Schrödinger equation for the  EMBED Equation.3 . 3. The Angular Solutions  EMBED Equation.3  We need to solve for Y and evaluate the C. a. Separate the angles Assume that  EMBED Equation.3 . Substitute into the differential equation and divide both sides by Y.  EMBED Equation.3  As usual, the two terms must separately be equal to a constant,  EMBED Equation.3 .  EMBED Equation.3  It can be shown that in fact  EMBED Equation.3  and  EMBED Equation.3 . This done is the following paragraph b, which may be skipped if time is short. b. Polar coordinates We are using spherical polar coordinates at this point because in the Hydrogen atom the potential energy function is spherically symmetric. Let us write the angular momentum vector in terms of polar coordinates.  EMBED Equation.3  We will require also the operators  EMBED Equation.3 , so we need to write the angular unit vectors in terms of the Cartesian unit vectors, as well.  EMBED Equation.3  Now, substitute these into  EMBED Equation.3  and collect the  EMBED Equation.3  terms. We obtain the polar versions of the angular momentum operators.  EMBED Equation.3  In terms of these, the raising and lowering operators become in polar coordinates  EMBED Equation.3  Now,  EMBED Equation.3 , so evidently  EMBED Equation.3 . The  EMBED Equation.3  is also an eigenfunction of the operator  EMBED Equation.3 ,  EMBED Equation.3 . Substitute the polar forms for the operators, as well as  EMBED Equation.3 . After some algebra, we obtain  EMBED Equation.3  We divide through by  EMBED Equation.3 .  EMBED Equation.3  Take those derivatives on the left hand side, and after some more algebra,  EMBED Equation.3  That last line is identical to the  EMBED Equation.3 -part of the Schrödinger equation. We can identify that  EMBED Equation.3  and that  EMBED Equation.3 . c. Spherical harmonics We have  EMBED Equation.3 . What remains is to solve for the  EMBED Equation.3 . [Known as  EMBED Equation.3  in the preceding paragraph b.]  EMBED Equation.3  This is a “well known” differential equation, whose solutions are the Associated Legendre Functions,  EMBED Equation.3 . These are defined in turn in terms of the Legendre Polynomials,  EMBED Equation.3  , with  EMBED Equation.3 .  EMBED Equation.3  See Table 4.1 in the text. So, we do not have to solve for  EMBED Equation.3  all over from scratch. What is left to do is to normalize the angular solution.  EMBED Equation.3   EMBED Equation.3  This has been done already, so we just look up the result:  EMBED Equation.3  See Table 4.2 in the text. The  EMBED Equation.3  are called Spherical Harmonics, and comprise a complete set of orthogonal functions. That means that any function of  EMBED Equation.3  can be expanded as a sum of  EMBED Equation.3 . C. Radial Equation  EMBED Equation.3  Since we have m for the azimuthal, or magnetic, quantum number, we’ll start using me for the mass of the electron. In addition, we’ll find it convenient to make a change of variable, letting  EMBED Equation.3 .  EMBED Equation.3  Classically, we do the same thing when solving Newton’s “Law” for orbits in a gravitational field. 1. Potential Energy a. Coulomb For the Hydrogen atom, the force acting on the electron is the electrostatic Coulomb force: The potential energy function is  EMBED Equation.3 . The radial equation therefore becomes  EMBED Equation.3  b. Another change of variable To make it more confusing, but to simplify the writing, we’ll change the variable again. Firstly, let  EMBED Equation.3 . Secondly, let  EMBED Equation.3  and  EMBED Equation.3 . Now, the radial equation at least has a cleaner appearance:  EMBED Equation.3  2. Radial Solution The construction of a solution is broken into two parts. The first part is called the asymptotic behavior, referring to the solution at very large distance from the proton or very close to the proton. a. Asymptotic behavior i) as  EMBED Equation.3 , which has the solution  EMBED Equation.3 . ii) as  EMBED Equation.3 , which has the solution  EMBED Equation.3 . Therefore, we propose that the radial solution has the form  EMBED Equation.3 . We’ll plug this into the radial equation and assume that  EMBED Equation.3  is a power series in  EMBED Equation.3 . b. Series solution We’ll need  EMBED Equation.3  and  EMBED Equation.3 . In the radial equation, the common factors of  EMBED Equation.3  will divide out.  EMBED Equation.3  Collect the like terms.  EMBED Equation.3  At this point, we assume that  EMBED Equation.3  and substitute it into the differential equation.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The coefficient of each power of  EMBED Equation.3  must vanish separately.  EMBED Equation.3  Solve for  EMBED Equation.3 .  EMBED Equation.3  The first coefficient is determined by the normalization condition,  EMBED Equation.3  for the moment, and the rest follow from the formula (called a recursion relation) above. However, the series must terminate, else  EMBED Equation.3  will diverge at large  EMBED Equation.3 . That is, the series cannot be infinitely long, for physical reasons. There must be some maximum value of j such that  EMBED Equation.3 . In the recursion relation  EMBED Equation.3  We define  EMBED Equation.3  to be the principle quantum number, because it labels the discrete energy levels (through the  EMBED Equation.3 ). 3. Radial Wavefunctions Now, we work backward from  EMBED Equation.3  to  EMBED Equation.3 . We had set  EMBED Equation.3  and  EMBED Equation.3  and  EMBED Equation.3 .  EMBED Equation.3  a. Ground state wave function The ground state wave function is  EMBED Equation.3  Here, the  EMBED Equation.3  is the Bohr Radius. To evaluate the  EMBED Equation.3  we normalize the R10.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  b. Excited state wave functions For  EMBED Equation.3 ,  EMBED Equation.3  and  EMBED Equation.3 . The  EMBED Equation.3  is a polynomial of degree  EMBED Equation.3  (that is,  EMBED Equation.3 ) whose coefficients are given by the recursion relation  EMBED Equation.3  and  EMBED Equation.3  is determined by normalizing the function. As they have been defined, the  EMBED Equation.3  are also “well known” as the Laguerre and Associated Laguerre Polynomials:  EMBED Equation.3  For us,  EMBED Equation.3 . After all the normalization, we end up with the wave functions for the energy states of the electron in the Hydrogen atoms:  EMBED Equation.3  See Tables 4.5, 4.6, and 4.7 as well as Figure 4.4 in the text. c. Energy levels The  EMBED Equation.3  is in the total energy, E:  EMBED Equation.3 .  EMBED Equation.3  These are exactly the same energy levels obtained for the classical Bohr model of the Hydrogen atom. However, Bohr’s model does not have  EMBED Equation.3 . The ground state wave function is  EMBED Equation.3  The corresponding ground state energy is E1 = -13.6 eV. The excited states have the energies  EMBED Equation.3 .  d. Degeneracy Recall that for any  EMBED Equation.3 ,  EMBED Equation.3  and  EMBED Equation.3 . Consequently, there may be several states whose energy is the same. This condition is called degeneracy. The energy levels are said to be degenerate. For instance, let  EMBED Equation.3 . Then there are three possible values of  EMBED Equation.3 . For each value of  EMBED Equation.3 , there are  EMBED Equation.3  possible values of  EMBED Equation.3 . Therefore, the degeneracy of the  EMBED Equation.3  level is 1 + 3 + 5 = 9. There are nine wave functions that have the same energy,  EMBED Equation.3 . e. Unbound states If  EMBED Equation.3 , then  EMBED Equation.3  and is real. But if  EMBED Equation.3 , then  EMBED Equation.3  is imaginary and we obtain free-particle oscillating solutions, with continuous energy values rather than discrete. We speak of the continuum of energy states lying above the discrete bound states. D. Spin A property of subatomic particles that Bohr did not put into his model is something called intrinsic spin. This is a built-in property of particles, just as is mass or electric charge. It is manifest in the magnetic moment a subatomic particle has, even neutrons. Recall the Stern–Gerlach experiment. 1. Spin Angular Momentum Operators We will define the spin operators by analogy with the orbital angular momentum. However, the eigenstates will not be functions of angles. a. Spin quantum numbers Define  EMBED Equation.3  such that  EMBED Equation.3  where  EMBED Equation.3  b. Commutation relations Just like the angular momentum operators,  EMBED Equation.3  2. Spin  EMBED Equation.3  For electrons,  EMBED Equation.3  a. Spinors Since  EMBED Equation.3 only, there are only two (2) eigenstates of  EMBED Equation.3  and  EMBED Equation.3 :  EMBED Equation.3 . These are sometimes called spinors.  EMBED Equation.3  A general spin state would expressed as a sum of spinors, but there is only two—  EMBED Equation.3 . If we measure the spin of the particle, we’ll get either  EMBED Equation.3  or  EMBED Equation.3 . The probability of getting  EMBED Equation.3  is  EMBED Equation.3 ,  EMBED Equation.3  is  EMBED Equation.3 . b. Spin operators The operators and spinors can be represented by matrices. Let the spinors be represented by column matrices:  EMBED Equation.3 . The operators are 2x2 square matrices, which can be figured out, knowing the eigenvalues. Suppose  EMBED Equation.3  and  EMBED Equation.3 . Carrying out the multiplications yields  EMBED Equation.3 . In a similar vein,  EMBED Equation.3 . There are also raising and lowering operators, such that  EMBED Equation.3 .  EMBED Equation.3 . III. Selected Quantum Mechanical Issues A. Vector Spaces 1. Properties A vector space consists of a set, or group, of mathematical entities, called vectors and scalars, which obey certain rules. a. Addition The sum of any two vectors is also a vector. That is, the sum of two members of the group is another member of the same group.  EMBED Equation.3  Vector addition obeys the associative law,  EMBED Equation.3 , and is communtative  EMBED Equation.3  b. Scalar multiplication The product of a scalar and a vector is also a vector in the vector space. Scalar multiplication is associative and distributive.  EMBED Equation.3  c. Null vector & inverse vector There exists a null vector, such that  EMBED Equation.3  . For every vector, there exists an inverse vector, such that  EMBED Equation.3 . d. Basis vectors & components A vector in the vector space it is said to be linearly independent of a set of other vectors if it cannot be written as a linear combination of that set of vectors. A basis set is a collection of linearly independent vectors that spans the vector space. That means that every vector can be written as a linear combination of the members of this set. The dimension of the vector space is the number of vectors in the basis set, namely n. Given a set of basis vectors,  EMBED Equation.3 , an arbitrary vector can be written as  EMBED Equation.3 . The {ai} are said to be components of the vector  EMBED Equation.3  with respect to the basis  EMBED Equation.3 . 2. Inner Products a. Properties The inner product of two vectors  EMBED Equation.3  and  EMBED Equation.3  is a complex number, such that:  EMBED Equation.3  b. Orthonormal vectors A unit vector is a vector whose norm is equal to one:  EMBED Equation.3  On the other hand, two vectors are orthogonal if their inner product is zero:  EMBED Equation.3  A set of mutually orthogonal, and normalized, vectors is called an orthonormal set.  EMBED Equation.3  c. Schwarz inequality  EMBED Equation.3  3. Linear Transformations A transformation,  EMBED Equation.3 , acts on a vector in the vector space to produce another vector.  EMBED Equation.3  In the case of linear transformations,  EMBED Equation.3  a. Matrix representation Let  EMBED Equation.3  be some arbitrary vector (in the vector space).  EMBED Equation.3 . Further, let a linear transformation be defined by  EMBED Equation.3 , (j = 1, 2, 3, 4, . . ., n).  EMBED Equation.3  Parenthetically, there is a form of notation in which the summation signs are not used; the summation is implied by a repeated index. I.e.,  EMBED Equation.3  The elements of  EMBED Equation.3  are defined with respect to the orthonormal basis set:  EMBED Equation.3 . A linear transformation acting on a vector can be represented by the multiplication of a square matrix and a column vector (a matrix with one column), thusly  EMBED Equation.3 . b. Doing things to matrices transpose complex conjugate hermitian conjugate commutator inverse matrix unitary matrix similarity transformation 4. Eigenvectors a. Eigen value equations  EMBED Equation.3 , where  EMBED Equation.3  is a complex number, an eigen value. The  EMBED Equation.3  is an eigen vector, or eigen function or eigen state of the operator,  EMBED Equation.3 . Commonly, the transformation is known, and the tasks are to solve for the eigen values and the eigen vectors. As illustration, consider the spin operator,  EMBED Equation.3 .  EMBED Equation.3  The equation  EMBED Equation.3  is called the characteristic equation. Its roots are the eigen values of the operator in the equation written in matrix form. The eigen vectors are obtained by plugging in the known eigen values.  EMBED Equation.3  b. Hermitian transformations B. Formal Quantum Mechanics 1. Postulates of Quantum Mechanics a. One For a system consisting of a particle in a conservative field there is an associated wavefunction,  EMBED Equation.3 . This wavefunction determines everything that can be known about the system, and is single valued in coordinates and time. In general, it is a complex function, and may be multiplied by a complex number without changing its physical significance. b. Two A Hermitian operator is associated with every physical observable. The eigen vectors of such an operator form a complete orthonormal basis set in Hilbert space. An arbitrary wavefunction can be expanded as a linear combination of eigen vectors. c. Three The development of the wavefunction with time, assuming the system is undisturbed, is governed by the Schrödinger Equation.  EMBED Equation.3  2. Statistical Interpretation a. Measurement Let  EMBED Equation.3  be the Hermitian operator corresponding to an observable physical quantity, Q. The eigen value equation for that operator would be  EMBED Equation.3 . Now consider a particle in a state  EMBED Equation.3 .  EMBED Equation.3  If a measurement of Q is made, one of the eigen values, qn will be the result. The probability obtaining a particular qn is  EMBED Equation.3 , where  EMBED Equation.3 . When the measurement is made, the wavefunction collapses to the eigen function,  EMBED Equation.3 . If the measurement were immediately repeated, the same qn.would be the result. The expectation value of Q is the weighted sum over all the eigenvalues.  EMBED Equation.3  This is the expected average of measurements performed on many independent identical systems. 3. 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