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Chapter 7: Q9CQ (page 278)

Knowing precisely all components of a nonzero L鈫�would violate the uncertainty principle, but knowingthat L鈫�is precisely zerodoes not. Why not?

(Hint:For l=0 states, the momentum vector p鈫� is radial.)

Short Answer

Expert verified

Knowingthat L鈫�is precisely zerodoes not violate uncertainty principle because the components commute in this case making them measurable simultaneously.

Step by step solution

01

Angular momentum commutation relation

The angular momentum components follow the commutation relation

Li,Lj=i鈩�LkLj,Lk=i鈩�LiLk,Li=i鈩�Lj ..... (I)

Here, 鈩� is the reduced Planck's constant.

02

Step 2:Determining the commutation for zero angular momentum

Observables which do not commute cannot be measured simultaneously. Thus it is evident from equation (I) that components of angular momentum cannot be measured simultaneously. But if all the components are measured to be zero, the right hand sides of equations (I) become zero and the components commute. Thus simultaneous measurement of angular momentum components if the values are zero does not violate uncertainty principle.

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Most popular questions from this chapter

In Appendix G. the operator for the square of the angular momentum is shown to be

L^2=-h2[csc胃鈭�鈭�胃sin胃鈭�鈭�胃+csc2胃鈭�2鈭�蠒2]

Use this to rewrite equation (7-19) asL^2鈿�桅=-Ch2鈿�桅

Verify that the solution given in equation (7.6) satisfy differential equations (7.5) as well as the required boundary conditions.

Taking then=3states as representative, explain the relationship between the complexity numbers of nodes and antinodes-of hydrogen's standing waves in the radial direction and their complexity in the angular direction at a given value of n. Is it a direct or inverse relationship, and why?

An electron is in anI = 3state of the hydrogen atom, what possible angles might the angular momentum vector make with the z-axis.

Using the functions given in Table 7.4, verify that for the more circular electron orbit in hydrogen (i.e.,l=n-1), the radial probability is of the form

P(r)鈭�r2ne-2r/nao

Show that the most probable radius is given by

rmostprobable=n2ao

See all solutions

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