bei48482_FM

Exercises 197

EXERCISES

Press on, and faith will catch up with you. — Jean D’Alembert

5.1 Quantum Mechanics

1. Which of the wave functions in Fig. 5.15 cannot have physical
   significance in the interval shown? Why not?


2. Which of the wave functions in Fig. 5.16 cannot have physical
   significance in the interval shown? Why not?
   3. Which of the following wave functions cannot be solutions of
   Schrödinger’s equation for all values of x? Why not? (a) 
   Asecx; (b) Atanx; (c) Aex
   2
   ; (d) Aex
   2
   .
   4. Find the value of the normalization constant Afor the wave
   function Axex

(^2)  2
.

5. The wave function of a certain particle is Acos^2 xfor
    2 x2. (a) Find the value ofA. (b) Find the proba-
   bility that the particle be found between x0 and x  4.

5.2 The Wave Equation

6. The formula yAcos (txν), as we saw in Sec. 3.3, de-
   scribes a wave that moves in the xdirection along a stretched
   string. Show that this formula is a solution of the wave equa-
   tion, Eq.(5.3).


7. As mentioned in Sec. 5.1, in order to give physically meaning-
   ful results in calculations a wave function and its partial deriva-
   tives must be finite, continuous, and single-valued, and in addi-
   tion must be normalizable. Equation (5.9) gives the wave
   function of a particle moving freely (that is, with no forces
   acting on it) in the xdirection as
   Ae(i)(Etpx)
   where Eis the particle’s total energy and pis its momentum.
   Does this wave function meet all the above requirements? If
   not, could a linear superposition of such wave functions meet
   these requirements? What is the significance of such a superpo-
   sition of wave functions?

5.4 Linearity and Superposition

8. Prove that Schrödinger’s equation is linear by showing that
   a 1  1 (x, t)a 2  2 (x, t)
   is also a solution of Eq. (5.14) if  1 and  2 are themselves
   solutions.

5.6 Operators

9. Show that the expectation values pxand xpare related by

pxxp

This result is described by saying that pand xdo not commute

and it is intimately related to the uncertainty principle.

10. An eigenfunction of the operator d^2 dx^2 is sinnx, where n
    1, 2, 3,.... Find the corresponding eigenvalues.



i

(a)(b)(c)

(d)(e)(f)



x



x



x



x



x



x

Figure 5.15

(a)(b) (c)



(d)(e) (f)



x



x x



x



x



x

Figure 5.16

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