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Chapter 7: Q7CQ (page 278)

Mathematically equation (7-22) is the same differential equation as we had for a particle in a box-the function and its second derivative are proportional. But(蠒)for m1= 0is a constant and is allowed, whereas such a constant wave function is not allowed for a particle in a box. What physics accounts for this difference?

Short Answer

Expert verified

The wave function m1= 0, which has no dependence and satisfies the condition without forcing anything to be zero everywhere.

Step by step solution

01

Significance of quantum numbers

The quantum number is used to describe the trajectory as well as the movement of the electron in an atom. According to the Pauli Exclusion principal, no electron in an atom can have same set of the quantum numbers.

02

Explanation of physics accounts for the difference

In case of the hydrogen atom and other particles in the spherical well, no barrier is encountered by varying the azimuthal angle over the complete range of values from 0 to 2. The wave function is not required to be zero that is encountered and the requirement instead is only that wave function at any angle must return to the same value at an angle that is greater. The wave function for m1 = 0, which has no dependence, already satisfies the condition without forcing anything to be zero everywhere.

Therefore, the wave function m1 = 0 , which has no dependence, and satisfies the condition without forcing anything to be zero everywhere.

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Most popular questions from this chapter

The Diatomic Molecule: Exercise 80 discusses the idea of reduced mass. Classically or quantum mechanically, we can digest the behavior of a two-particle system into motion of the center of mass and motion relative to the center of mass. Our interest here is the relative motion, which becomes a one-particle problem if we merely use 渭for the mass for that particle. Given this simplification, the quantum-mechanical results we have learned go a long way toward describing the diatomic molecule. To a good approximation, the force between the bound atoms is like an ideal spring whose potential energy is 12kx2, where x is the deviation of the atomic separation from its equilibrium value, which we designate with an a. Thus,x=r-a . Because the force is always along the line connecting the two atoms, it is a central force, so the angular parts of the Schr枚dinger equation are exactly as for hydrogen, (a) In the remaining radial equation (7- 30), insert the potential energy 12kx2and replace the electron massm with 渭. Then, with the definition.f(r)=rR(r), show that it can be rewritten as

-魔22渭d2dr2f(r)+魔2I(I+1)2渭r2f(r)+12kx2f(r)=Ef(r)

With the further definition show that this becomes

-魔22渭d2dx2g(x)+魔2I(I+1)2渭(x+a)g(x)+12kx2g(x)=Eg(x)

(b) Assume, as is quite often the case, that the deviation of the atoms from their equilibrium separation is very small compared to that separation鈥攖hat is,x<<a. Show that your result from part (a) can be rearranged into a rather familiar- form, from which it follows that
E=(n+12)魔k渭+魔2I(I+1)2渭a2n=0,1,2,...I=0,1,2,...

(c)

Identify what each of the two terms represents physically.

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

(a) For one-dimensional particle in a box, what is the meaning of n? Specifically, what does knowing n tell us? (b) What is the meaning of n for a hydrogen atom? (c) For a hydrogen atom. What is the meaning of landml?

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.)

For an electron in the(n,l,ml)=(2,0,0) state in a hydrogen atom, (a) write the solution of the time-independent Schrodinger equation,

(b) verify explicitly that it is a solution with the expected angular momentum and energy.

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