FIG.13:Conservation of probability We take again the free particle as an example.The wave function propagating along =er-wt) leads immediately to a constant probability density p =1=const. The current density j=（e0-e0Te0） n光w-Xo. The velocity of classical particle v is just the current density and we have V.j=0 which is consistent with the result op/ot=0. We can extend the region of interest to the entire space.In this case the wave function must go to zero in the infinity-otherwise it would not be normalizable,so must the current mc,0=0,m。j=0 We thus have p(r,t)dr=0 which means the conservation of total probability in the entire space. C.Stationary Schrodinger equation For many physical models the potential V(r)is independent of time t.In those cases the Schrodinger Equation (we denote the time-dependent wave function by and stationary solution byin the following) n2=-会a0+ve0 t 31
FIG. 13: Conservation of probability. We take again the free particle as an example. The wave function propagating along x ψ = e i(kx−ωt) leads immediately to a constant probability density ρ = 1 = const. The current density j = − i~ 2m ¡ e −i(kx−ωt)∇e i(kx−ωt) − e i(kx−ωt)∇e −i(kx−ωt) ¢ = − i~ 2m 2ikx0 = ~k m x0 = vx0. The velocity of classical particle v is just the current density and we have ∇ · j = 0 which is consistent with the result ∂ρ/∂t = 0. We can extend the region of interest to the entire space. In this case the wave function ψ must go to zero in the infinity - otherwise it would not be normalizable, so must the current j lim r→±∞ ψ(r, t) = 0, lim r→±∞ j = 0 We thus have d dt ZZZ +∞ −∞ ρ(r, t)d 3 r = 0 which means the conservation of total probability in the entire space. C. Stationary Schr¨odinger equation For many physical models the potential V (r) is independent of time t. In those cases the Schr¨odinger Equation (we denote the time-dependent wave function by Ψ and stationary solution by ψ in the following) i~ ∂Ψ(r, t) ∂t = − ~ 2 2m ∇2Ψ(r, t) + V (r)Ψ(r, t) 31
Xo FIG.14:Example:Free particle can be solved by the method of separation of variables(the physicist's first line of attack on any partial differential equation):We look for solutions that are simple products. 亚(r,t)=(r)f(t) where (r)(lower-case)is a function of r alone,and f(t)is a function of t alone.On its face,this is an absurd restriction,and we cannot hope to get more than a tiny subset of all solutions in this way.But hang on,because the solutions we do obtain turn out to be of great interest.Moreover,as is typically the case with separation of variables,we will be able at the end to patch together the separable solutions in such a way as to construct the most general solution. The Schrodinger Equation reads h品间={(p2+vo）o}r间 Or dividing through by(r)f(t) h四_（-然2+V)回) (r) Now the left side is a function of t alone,and the right side is a function of r alone.The only way this can possibly be true is if both sides are in fact constant-otherwise,by varying t,I could change the left side without touching the right side,and the two would no longer be equal.The separation constant E is introduced in the process of separation of variable and recognized to be the energy of the system.Then we have h品10=0 32
j x0 FIG. 14: Example: Free particle can be solved by the method of separation of variables (the physicist’s first line of attack on any partial differential equation): We look for solutions that are simple products, Ψ(r, t) = ψ(r)f(t) where ψ(r) (lower -case) is a function of r alone, and f(t) is a function of t alone. On its face, this is an absurd restriction, and we cannot hope to get more than a tiny subset of all solutions in this way. But hang on, because the solutions we do obtain turn out to be of great interest. Moreover, as is typically the case with separation of variables, we will be able at the end to patch together the separable solutions in such a way as to construct the most general solution. The Schr¨odinger Equation reads · i~ d dtf(t) ¸ ψ(r) = ½µ− ~ 2 2m ∇2 + V (r) ¶ ψ(r) ¾ f(t) Or dividing through by ψ(r)f(t) i~ d dtf(t) f(t) = ³ − ~ 2 2m∇2 + V (r) ´ ψ(r) ψ(r) Now the left side is a function of t alone, and the right side is a function of r alone. The only way this can possibly be true is if both sides are in fact constant - otherwise, by varying t, I could change the left side without touching the right side, and the two would no longer be equal. The separation constant E is introduced in the process of separation of variable and recognized to be the energy of the system. Then we have i~ ∂ ∂tf(t) = f(t)E 32
with solution f(t)=e and (v)) This is the time-independent Schrodinger Equation,or Stationary Schrodinger Equation.The natural boundary conditions should be imposed on:first of all the wave function should be single-valued and limited.Furthermore(rand its spatial derivations should be continuous(there may be exceptions in the above conditions for some artificially simplified V(r)) The following chapters will be devoted to solving the time-independent Schrodinger equa- tion,for a variety of simple potentials.But before we get to that you have every right to ask: What's so great about separable solutions? After all,most solutions to the (time-dependent)Schrodinger equation do not take the form(r)f(t).I offer three answers-two of them physical and one mathematical (i)They are stationary states.Although the wave function itself,(r,t)=(r)e does (obviously)depend on t,the probability density(r,t)2=(r)2does not-the time dependence cancels out.The same thing happens in calculating the expectation value of any dynamical variable;Equation(3)reduces to Summarized in one word,stationary state is stable to all the test,i.e.the result doesn't change with time.If a particle is initially put in a state with energy E,it will stay there all the time,i.e. Ψ(r,0)=E(r)→Ψ（红，）=Er)eE and moreover pr,)=(（E(r)et）'ere =(r)E(r)=p(r) 33
with solution f(t) = e − i ~ Et and µ − ~ 2 2m ∇2 + V (r) ¶ ψ(r) = Eψ(r) This is the time-independent Schr¨odinger Equation, or Stationary Schr¨odinger Equation. The natural boundary conditions should be imposed on: first of all the wave function should be single-valued and limited. Furthermore ψ(r) and its spatial derivations should be continuous (there may be exceptions in the above conditions for some artificially simplified V (r)) The following chapters will be devoted to solving the time-independent Schr¨odinger equation, for a variety of simple potentials. But before we get to that you have every right to ask: What’s so great about separable solutions? After all, most solutions to the (time-dependent) Schr¨odinger equation do not take the form ψ(r)f(t). I offer three answers – two of them physical and one mathematical: (i) They are stationary states. Although the wave function itself, Ψ(r, t) = ψ(r)e − i ~ Et does (obviously) depend on t, the probability density |Ψ(r, t)| 2 = |ψ(r)| 2 does not - the time dependence cancels out. The same thing happens in calculating the expectation value of any dynamical variable; Equation (3) reduces to hQ (x, p)i = Z +∞ −∞ dxψ∗ (x) Q µ x, −i~ ∂ ∂x¶ ψ (x) Summarized in one word, stationary state is stable to all the test, i.e. the result doesn’t change with time. If a particle is initially put in a state with energy E, it will stay there all the time, i.e. Ψ(r, 0) = ψE(r) ⇒ Ψ(r, t) = ψE(r)e − i ~ Et and moreover ρ(r, t) = ³ ψE(r)e − i ~ Et´∗ ψE(r)e − i ~ Et = ψ ∗ E(r)ψE(r) = ρ(r) 33
a02{r(ge词】 -e(eev(r)e）j ,)=f(eoe)(-ih品)(ee-）r -/ao(-h品）sor Problem5 Show that the erpectation value of the velocity is equal to the time derivative of the erpectation value of position 问-号 And it is customary to work with momentum (p=mv) For stationary states,every expectation value is constant in time.In particular,(r)is constant,and hence(p)=0.Nothing ever happens in a stationary state.(page 16 Griffiths) (ii)They are states of definite total energy.In classical mechanics,the total energy(kinetic plus potential)is called the Hamiltonian: ù2 ,刊=+v回 The corresponding Hamiltonian operator,obtained by the canonical substitution p -ih是，is therefore Thus the time-independent Schrodinger equation can be written as the eigenvalue equation of operatorH Aw=E沙 which is the same as what we met in the introduction.And the expectation value of the total energy is (H)=v*dx=Eldr =Edr =E 34
j(r, t) = − i~ 2m ψ ∗ E(r)e i ~ Et∇ ³ ψE(r)e − i ~ Et´ −ψE(r)e − i ~ Et∇ ³ ψ ∗ E(r)e i ~ Et´ = − i~ 2m {ψ ∗ E(r)∇ψE(r) − ψE(r)∇ψ ∗ E(r)} = j(r) hpxi = ZZZ ³ ψ ∗ E(r)e i ~ Et´ µ −i~ ∂ ∂x¶ ³ ψE(r)e − i ~ Et´ d 3 r = ZZZ ψ ∗ E(r) µ −i~ ∂ ∂x¶ ψE(r)d 3 r Problem 5 Show that the expectation value of the velocity is equal to the time derivative of the expectation value of position hvi = d hxi dt And it is customary to work with momentum (p = mv) hpi = m d hxi dt For stationary states, every expectation value is constant in time. In particular, hxi is constant, and hence hpi = 0. Nothing ever happens in a stationary state. (page 16 Griffiths) (ii) They are states of definite total energy. In classical mechanics, the total energy (kinetic plus potential) is called the Hamiltonian: H(x, p) = p 2 2m + V (x) The corresponding Hamiltonian operator, obtained by the canonical substitution p → −i~ ∂ ∂x , is therefore Hˆ = − ~ 2 2m ∂ 2 ∂x2 + V (x) Thus the time-independent Schr¨odinger equation can be written as the eigenvalue equation of operator Hˆ Hψˆ = Eψ which is the same as what we met in the introduction. And the expectation value of the total energy is hHi = Z ψ ∗Hψdx ˆ = E Z |ψ| 2 dx = E Z |Ψ| 2 dx = E 34
(Notice that the normalization of entails the normalization of .Moreover, 2=(fv=(Ev)=E(i)=E2 and hence )=p㎡ed=2Pd在=E2 So the variance of H is 扇=(H2〉-(H)2=E2-E2=0 But remember,if o=0,then every member of the sample must share the same value (the distribution has zero spread).Conclusion:A separable solution has the property that every measurement of the total energy is certain to return the value E.(That's why I chose that letter for the separation constant.) (iii)The general solution is a linear combination of separable solutions.As we're about to discover,the stationary Schrodinger equation yields an infinite collection of solu- tions,each with its associated value of the separation constant (r),2(c),的(r) E1,E2,E… thus there is a different wave function for each allowed energy 亚1(c,t)=1(）)eBh 业2(c,)=(r)e-iB/h,… Superposition of two or more states with different energies is not a stationary state. A much more general solution can be constructed by a linear combination of the separable solutions 重任，)=∑snge-n n=1 which means essentially the principle of superposition of states.The main point is this:Once you've solved the time-independent Schrodinger equation,you're essentially done;getting from them to the general solution of the time-dependent Schrodinger equation is simple and straightforward. 35
(Notice that the normalization of Ψ entails the normalization of ψ.) Moreover, Hˆ 2ψ = Hˆ ³ Hψˆ ´ = Hˆ (Eψ) = E ³ Hψˆ ´ = E 2ψ and hence H 2 ® = Z ψ ∗Hˆ 2ψdx = E 2 Z |ψ| 2 dx = E 2 So the variance of H is σ 2 H = H 2 ® − hHi 2 = E 2 − E 2 = 0 But remember, if σ = 0, then every member of the sample must share the same value (the distribution has zero spread). Conclusion: A separable solution has the property that every measurement of the total energy is certain to return the value E . (That’s why I chose that letter for the separation constant.) (iii) The general solution is a linear combination of separable solutions. As we’re about to discover, the stationary Schr¨odinger equation yields an infinite collection of solutions, each with its associated value of the separation constant ψ1 (r), ψ2 (r), ψ3 (r)· · · E1, E2, E3 · · · thus there is a different wave function for each allowed energy Ψ1 (r, t) = ψ1 (r) e −iE1t/~ , Ψ2 (r, t) = ψ2 (r) e −iE2t/~ , · · · Superposition of two or more states with different energies is not a stationary state. A much more general solution can be constructed by a linear combination of the separable solutions Ψ (r, t) = X∞ n=1 cnψn (r) e −iEnt/~ which means essentially the principle of superposition of states. The main point is this: Once you’ve solved the time-independent Schr¨odinger equation, you’re essentially done; getting from them to the general solution of the time-dependent Schr¨odinger equation is simple and straightforward. 35