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1、1,Outline,2,概况1您的内容打在这里，或者通过复制您的文本后。概况2您的内容打在这里，或者通过复制您的文本后。概况3您的内容打在这里，或者通过复制您的文本后。,+,+,+,整体概况,3,In Chapter 1, we have studied a lot about the wave function and how you use it to calculate various quantities of interest.,Question: How do you get (x,t) in the first place ? How do you go about solvin
2、g the Schrdinger equation ?,4,In this Chapter，we assume that the potential (or potential energy function), V(x,t)=V(x), of the system, is independent of time t !,5,Now the left side is a function of t alone, and the right side is a function of x alone.,6,The only way this can be possibly be true is
3、if both sides are in fact constant, we shall call the separation constant E. Then,Separation of variables has turned a partial differential equation into two ordinary differential equations (Eq. 2.3 and 2.4).,7,The first equation 2.3 is easy to solve, the general solution of Eq.2.3 is,The second equ
4、ation 2.4 is called the time -independent Schrdinger equation, we can go on further with it until the potential V(x) is specified.,The rest of this chapter will be devoted to solving the time-independent Schrdinger equation 2.4, for a variety of simple potentials. But before we get to that we would
5、like to consider further the question:,8,Whats so great about separable solution ? 可分离的解（即 (x,t)=(x) f(t) ）为何如此重要？,After all, most solutions to the (time-dependent) Schrdinger equation do not take the form (x)f(t). We will offer three answers two of them physical and one mathematical:,9,NOTE: for no
6、rmalizable solutions, E must be real (see Problem 2.la).,10,Nothing ever happens in the Stationary State (x,t) !,11,12,Please distinguish the operator with “hat” () to its dynamical variable in Eq.2.12.,13,Conclusion: A separable solution has the property that every measurement of the total energy i
7、s certain to return the value E. (Thats why we chose that letter E for the separation constant.),3. The general solution is a linear combination of separable solutions,14,Now the (time-dependent) Schrdinger equation (Eq. 1.1) has the property that any linear combination5 of solutions is itself a sol
8、ution.,15,It so happens that every solution to the (time -dependent) Schrdinger equation can be written in this form it is simply a matter of finding the right constants (c1, c2, c3, c4, ）so as to fit the initial conditions for the problem at hand.,Once we have found the separable solutions, then, w
9、e can immediately construct a much more general solution, of the form,16,Youll see in the following sections how all this works out in practice, and in Chapter 3 well put it into more elegant language, but the main point is this: Once youve solved the time-independent Schrdinger equation, youre esse
10、ntially done; getting from there to the general solution of the time-dependent Schrdinger equation is simple and straightforward.,17,Brief summary of section 2.1,V(x,t)=V(x),boundary conditions,18,19,The first example to solve the Schrdinger equation is the infinite square well,20,A particle in this
11、 potential is completely free, except at the two ends (x = 0 and x = a), where an infinite force prevents it from escaping.,Outside the well, (x,t)=0 (the probability of finding the particle there is zero). Inside the well, where V=0, the time-independent Schrdinger equation (Equation 2.4) reads,21,
12、Equation 2.17 is the (classical) simple harmonic oscillator equation; the general solution is,Typically, these constants are fixed by the boundary conditions of the problem. What are the appropriate boundary conditions for (x)?,22,For the infinite square well, both (x) and d(x)/dx are continuous at
13、the two ends (x = 0 and x = a) !NOTE: only the first condition of these is applied since the potential goes to infinity here!,Continuity of (x) requires that,23,24,Curiously, the boundary condition at x = a does not determine the constant A, but rather the constant k, and hence the possible values o
14、f E can be obtained from Eq.2.17 and 2.22:,In sharp contrast to the classical case, a quantum particle in the infinite square well cannot have just any old energy only these special allowed values.,25,As promised, the time-independent Schrdinger equation has delivered an infinite set of solutions, o
15、ne for each integer n. The first few of these are plotted in Figure 2.2:,26,they look just like the standing waves on a string of length a. 1, which carries the lowest energy, is called the ground state; the others, whose energies increase in proportion to n2, are called excited states.,27,The wave
16、functions n(x) have some interesting and important properties:,28,29,Note that: this argument does not work if mn (can you spot the point at which it fails?); in that case normalization tells us that the integral is 1.,In fact, we can combine orthogonality and normalization into a single statement :
17、,We say that the s are orthonormal.,30,4. They are complete,In the sense that any other function, f ( x ), can be expressed as a linear combination of them n(x) :,“Any” function can be cxpanded in this way is sometimes called Dirichlets (狄利克雷) theorem.,The expansion coefficients (cn) can be evaluate
18、d for a given f (x) - by a method called Fouriers trick(技巧), which beautifully exploits the orthonormality of n(x) : Multiply both sides of Equation 2.28 by m*(x) , and integrate.,31,Thus the mth coefficient in the expansion of f (x) is given by,32,These four properties are extremely powerful, and t
19、hey are not peculiar(特有的) to the infinite square well. The first is true whenever the potential itself is an even function; The second is universal, regardless of the shape of the potential. Orthogonality is also quite general - we1 show you the Proof in Chapter 3. Completeness holds for all the pot
20、entials you are likely to encounter, but the Proofs tend to be nasty and laborious; Im afraid most physicists simply assume completeness and hope for the best.,33,The stationary states (Equation 2.6) for the infinite square well are evidently by using of Eq. 2.23 and 2.24 :,The most general solution
21、 to the (time-dependent) Schrdinger equation is a linear combination of stationary states Eq.2.31:,34,In general, when t = 0, according to Equation 2.32, we can fit any prescribed initial wave function, (x,0), by appropriate choice of the coefficients cn:,The completeness of the (x,0)s (confirmed in
22、 this case by Dirichlets theorem) guarantees that we can always express (x,0) in this way, and their orthonormality licenses the use of Fouriers trick to determine the actual coefficients. For example, infinite square well, we have,35,Given the initial wave function, (x,0),We first compute the expan
23、sion coefficients cn，by using of Equation 2.33,Then plug these into Equation 2.32 to obtain (x,t),Armed with the wave function, we are in a position to compute any dynamical quantities of interest, using the procedures in Chapter 1.,And this same ritual applies to any potentialthe only things that c
24、hange are the functional form of the s and the equation for the allowed energies.,36,Homework: Example 2.1, Example 2.2 Problem 2.7, Problem 2.37,37,The paradigm for a classical harmonic oscillator is a mass m attached to a spring (弹力) of force constant k. The motion is governed by Hookes law:,38,Of
25、 course, theres no such thing as a perfect simple harmonic oscillator if stretch it too far the spring is going to break, and typically Hookes law fails long before that point is reached.,But practically any potential is approximately parabolic(抛物线的), in the neighborhood of a local minimum (Figure 2
26、.3).,39,40,Thats why the simple harmonic oscillator is so important: Virtually(实际上; 事实上) any oscillatory motion is approximately simple harmonic, as long as the amplitude is small.,41,In the literature you will find two entirely different approaches to this problem.,The Quantum problem is to solve t
27、he Schrdinger equation for the potential,As we have seen, it suffices to solve the time independent Schrdinger equation:,42,The second is a diabolically(魔鬼似地) clever(聪明的) algebraic(代数的) technique, using so-called ladder(阶梯) operators. We will study the algebraic method firstly, because it is quicker
28、 and simpler (and more fun).,The first is a straightforward “brute force”(强力) solution to the differential equation, using the method of power series expansion; it has the virtue(优点) that the same strategy(策略) can be applied to many other potentials (in fact, we will use it in Chapter 4 to treat the
29、 Coulomb potential).,43,The idea is to factor the term in square brackets. If these were numbers easy:,To begin with, lets rewrite Equation 2.39 in a more suggestive form:,Here, however, its not quite so simple, because u and v are operators, and operators do not, in general, commute (i.e. uvvu ). S
30、till, this does invite us to take a look at the expressions, from Eq.2.40, we assume,44,Warning: Operators can be slippery to work with in the abstract, and you are bound to make mistakes unless you give them a “test function”, f (x), to act on. At the end you can throw away the test function, and y
31、oull be left with an equation involving the operators alone.,“product” 积,45,In the present case, we have,46,By using of Eq.2.42, the Schrdinger equation 2.40 becomes,Notice that the ordering of the factor a+ and a is important here !,47,The same argument, with a+ on the left, yields,Thus, Eq.2.42 Eq
32、.2.44, we have,48,Now, here comes the crucial (关键的) step:,49,Here, then, is a wonderful machine for grinding out new here solutions, with higher and lower energies if we can just find one solution, to get stared!,50,We called a ladder operators, because they allow us to climb up and down in energy;
33、a+ is called the raising operator, and a the lowering operator.,The “ladder” of states is illustrated in Figure 2.4.,51,But wait !,What if we apply the lowering operator a repeatedly ? Eventually were going to reach a state with energy less than zero , which (according to the general theorem in Prob
34、lem 2.2) does not exist !,At some point the machine must fail. How can that happen？,We know that a is a new solution to the Schrdinger equation, but there is no guarantee that it will be nonmalleable it might be zero, or its square integral might be infinite. Problem 2.11 rules out the latter possib
35、ility.,52,Conclusion: There must occur a “lowest rung(阶梯)” ( lets call it 0）such that,(see Eq.2.41),53,To determine the energy of this state, we plug it into the Schrdinger equation (in the form of Equation 2.46),54,With our foot now securely planted on the bottom rung (the ground state of the quant
36、um oscillator), we simply apply the raising operator to generate the excited state:,This method does not immediately determine the normalization factor An, which will be worked out by yourself in Problem 2.12.,55,We wouldnt want to calculate 50 in this way, but never mind: We have found all the allo
37、wed energies, and in principle we have determined the stationary states the rest is just computation.,56,Homework: Example 2.5,57,We return now to the Schrdinger equation for the harmonic oscillator (Equation 2.39):,We introduce the dimensionless variable,58,Our problem is to solve Equation 2.56, an
38、d in the process obtain the “allowed” values of K (and hence of E from Eq.2.57).,59,Note that the B term in Eq.2.59 is clearly not normalizable (it blows up as |x|); the physically acceptable solutions, then, have the asymptotic form,This suggests that we “peel off” the exponential part,60,so the Sc
39、hrdinger equation (Eq.2.56) becoms,61,62,This recursion(递推) formula is entirely equivalent to the Schrdinger equation itself.,From Eq.2.65: given a0 , Eq.2.65 enables us (in principle) to generate a2, a4, a6, given a1 , Eq.2.65 generates a3, a5, a7, . Let us write,63,Thus equation 2.65 determines h(
40、) in terms of two arbitrary (a0 and a1) which is just what we would expect, for a second-order differential equation.,However, not all the solutions so obtained are normalizable. For at very large j, the recursion formula 2.65 becomes (approximately),64,Now, if h goes like exp(2), then (remember ? t
41、hats what were trying to calculate) goes like exp(2/2) (Equation 2.61), which is precisely the asymptotic behavior we dont want.,65,There is only one way to wiggle out of this: For normalizable solutions the power series must terminate. There must occur some “highest” j (call it n) such that the rec
42、ursion formula spits out an+2 = 0 . Conclusion: the series hodd will be truncated at some highest n; the series heven must be zero from the start. For physically acceptable solutions, then, we must have,for some positive integer n, which is to say (referring to Equation 2.57) that the energy must be
43、 of the form,66,Thus, we recover, by a completely different method, the fundamental quantization condition we found algebraically in Equation 2.50.,(Eq.2.68 is obtained by putting K=2n+1 into Eq.65),67,68,In general, hn() will be a polynomial of degree n in , involving even powers only: if n is an e
44、ven integer, and odd powers only, if n is an odd integer. Apart from the overall factor (a0 or a1) they are the so-called Hermite polynomials, Hn().,The first few of them are listed in Table 2.1.,69,By tradition, the arbltrary multiplicative factor is chosen so that the coefficient of the highest po
45、wer of is 2n. With this convention, the normalized stationary states for the harmonic oscillator are,They are identical (of course) to the ones we obtained algebraically in Equation 2.50.,In Figure 2.5a we can plot n(x) for the first few ns.,70,71,The quantum oscillator is strikingly(醒目地) different
46、from its classical counterpart not only are the energies quantized, but the position distributions have some bizarre (怪诞的) features.,For instance: The probability of finding the particle outside the classically allowed range (that is, with x greater than the classical amplitude for the energy in que
47、stion) is not zero (see Problem 2.15). In all odd states the probability of finding the particle at the center of the potential well is zero. Only at relatively large n do we begin to see some resemblance(相似) to the classical case.,72,In Figure 2.5b, we have superimosed the classical position distri
48、bution on the quantum one (for n=100); if you smoothed out the bumps in the latter, the two would fit pretty well.,Fig.2.5 (b) Graph of |100|2, with the classical distribution superimposed.,73,74,We turn next to what should have been the simplest case of all: the free particle V(x)=0 everywhere.,As
49、youll see in a moment, the free particle free particle is in fact a surprisingly(令人惊讶地) subtle(难以捉摸的) and tricky(难处理的) example.,The time-independent Schrdinger equation reads,75,So far, its the same as inside the infinite square well (Equation 2.17), where the potential is also zero.,This time, howe
50、ver, we prefer to write the general solution of the free particle in exponential form (instead of sines and cosines) for reasons that will appear in due course:,Unlike the infinite square well, there are no boundary conditions to restrict the possible values of k (and hence of E as shown by Eq.2.75
