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204 C H A P T E R 6 The Schrodinger Equation
1925 and is now known as the Schrod inger equat ion. Like the classical wave equat ion, the Schrod inger equat ion relates the t ime and space derivatives of the wave function. T h e reasoning followed by Schrodinger is somewhat difficult and not important for o u r purposes . In any case, we can't der ive the Schrodinger equat ion jus t as we can't der ive Newton 's laws of motion. T h e validity of any fundamenta l equat ion lies in its a g r e e m e n t with exper iment . Al though it would be logical merely to postulate the Schrodinger equat ion, it is helpful to get some idea of what to expect by first consider ing the wave equation for pho tons , which is Equation 5-7 with speed v = c and with y(x,t) replaced bv the wave function for light, namely, the electric field %(x,t).
d2% dx2
1 d 2 %
c2 dt2 6-1 Classical wave equation
As discussed in Chap te r 5, a particularly impor tan t solution of this equat ion is the ha rmonic wave function %(x,t) = (?o cos (kx — <at). Differentiating this function twice we obtain d2%/dt2 = - w 2 g 0 c o s ( & c - tat) = -<02%(x,t) and d2%/dx2 = — k2%(x,t). Substitution into Equation 6-1 then gives
= - 4 c-
OT
(i) = kc 6-2
Using w = E/h and p = hk, we have
E = pc 6-3
which is the relat ion between the energv and m o m e n t u m of a pho ton .
Let us now use the de Broglie relations for a particle such as an electron to find the relation between o» and k for electrons which is analogous to Equat ion 6-2 for pho tons . We can then use this relat ion to work backwards and see how the wave equat ion for electrons must differ from Equation 6 -1 . T h e energy of a part icle of mass m is
2 m + V 6-4
where V is the potential energv. Using the de Broglie relations we obtain
E nergy-momen tu m relation for photon
h (D h2k2
2 m + V 6-5
Th i s differs from Equation 6-2 for a pho ton because it contains the potential energy V and because the angular frequency w does not vary linearly with k. Note that we get a factor of OJ when we differentiate a ha rmon ic wave function with respect to t ime
S E C T I O N 6-4 Expectation Values and Operators 217
T h e expectat ion value is the same as the average value of .v that we would expect to obtain from a m e a s u r e m e n t of the positions of a large n u m b e r of particles with the same wave function ty(x,<). As we have seen, for a particle in a state of definite energy the probability distr ibution is i n d e p e n d e n t of t ime. T h e expectation value is then given bv
•+x
(x) = xi/>*(x)t/»(v) dx 6-28 Expectation value J -00
From the infinite-square-well wave functions, we can see by sym-metrv (or bv direct calculation) that (x) is L/2, the midpoin t of the well.
T h e expectat ion value of any function / ( x ) is given by
</(*)> = I f{x)$*$dx 6-29
For example , (x2) can be calculated from the wave functions, above, for the infinite square well of width L. It is left as an exercise to show that for that case
L2 12
(x2) = V - d i " ! 6-30 3 2n IT
We should note that we don ' t necessarily expect to measu re the expectat ion value. For example , for even n , the probability of measur ing x in some r ange dx at t he midpoin t of the well x = L/2 is zero because the wave function sin (mrx/L) is zero there . We get (x) = L/2 because the probability function is symmetrical about that point.
Optional
Expectation value of momentum
Operators
If we knew the m o m e n t u m p of a particle as a function of x, we could calculate the expectat ion value (p) f rom Equation 6-29. However , it is impossible in principle to find p as a function of x since, according to the uncer ta inty principle, both p and x cannot be de t e rmined at the same t ime. T o find (p) we need to know the distr ibution function for m o m e n t u m , which is equivalent to the distr ibution function A(k) discussed in Section 5-4. As discussed there , if we know i/»(x) we can find A (k) and vice versa bv Four ier analysis. Fortunately we need not d o this each t ime. It can be shown from Four ier analysis that (p) can be found from
» - f > ( 7 5 i ) * * Similarly (/>*) can be found from
6-31
S E C T I O N 6-5 Transitions between Energy States 219
in te rms of position and m o m e n t u m and replace the m o m e n t u m variables by the app rop r i a t e opera to rs to obtain the Hamil tonian o p e r a t o r for the system.
Quest ions
4. For what kind of probability distr ibution would vou expect to get the expectat ion value in a single measurement?
5. Is ( x 2 ) the same as (x) 2 ?
6-5 Transitions between Energy States
We have seen that the Schrodinger equat ion leads to energy quant izat ion for b o u n d systems. T h e existence of these energy levels is de t e rmined experimental ly by observation of the energy emit ted o r absorbed when the system makes a transition from one level to ano the r . In this section we shall consider some aspects of these transit ions in one d imension. T h e results will be readily applicable to m o r e complicated situations.
In classical physics, a charged particle radiates when it is accelerated. If t he charge oscillates, the frequency of the radiation emit ted equals the frequency of oscillation. A stationary charge distr ibution does not radia te .
Consider a particle with charge q in a q u a n t u m state n described by the wave function
where En is the energy and <|/„(x) is a solution of the time-i n d e p e n d e n t Schrodinger equat ion for some potential energy V(x). T h e probability of finding the charge in dx is ^ f ^ n dx- If we make many measu remen t s on identical svstems (i.e., particles with the same wave function), the average a m o u n t of charge found in dx will be qtyfity n dx. We there fore identify q^t^n with the charge density p. As we have poin ted out , the probability density is i n d e p e n d e n t of t ime if the wave function contains a single energy, so the charge density for this state is also i n d e p e n d e n t of t ime:
pn = qV*(x,ty9n(x,t) = ?i|»*(xW/(x) = q^mt
We should therefore expect that this stationary charge distribution would n o t radiate. (This a rgumen t , in tin- case oi the hydrogen a tom, is the quantum-mechanica l explanat ion of Bohr 's postulate of nonrad ia t ing orbits.) However , we d o observe that systems m a k e transit ions from one energy state to ano the r with the emission or absorpt ion of radiat ion. T h e cause of the transit ion is the interaction of the electromagnetic field with
t T o simplify the notation in this section we shall sometimes omit the functional dependence and merely write tlin for 0„(x) and for y^x.t).