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Chapter 7: Q73E (page 283)

Verify for the angular solutions ⊙(θ)φ(ϕ)of Table 7.3 that replacing ϕ with ϕ+π and replacing θ with π-θgives the same function whenis even and the negative of the function when lis odd.

Short Answer

Expert verified

And for all the cases when l=0 and l=2 , either both or neither of them will change signs and hence the function will remain unchanged.

Step by step solution

01

Replacing ϕ with ϕ+π :

A function which acts as a mathematical description of a quantum state of an isolated quantum system, is called a wave function.

In Azimuthal wave function,φ(ϕ)=eimlϕ,

Where, ϕis the colatitude and m1is the magnetic quantum number.

By replacing ϕwith(ϕ+π) , you get,
role="math" localid="1659699420914" φ(ϕ+π)=eiml(ϕ+π)=eiml(π)=eimlϕ(cosmlπ+isinmlπ)

From the above equation, you get, sine term is zero, while cosine term is + 1 while is even and cosine term is -1 when it is odd.

Hence,φ(ϕ) the changes sign when is odd and remains changed otherwise.

02

Replacing θ with π-θ :

In the function ⊙(θ), here, θis the colatitude

By replacing θwithπ-θ you get,
cos(π-θ)=-cos(θ)sin(π-θ)=sin(θ)

Hence, only the terms having odd power of cos(θ)will change.

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Most popular questions from this chapter

If the constantCxinequation(7-5)were positive, the general mathematical solution would be

Ae+cxx+Be-cxx

Show that this function cannot be 0 at two points. This makes it an unacceptable solution for the infinite well, since it cannot be continuous with the wave functions outside the walls, which are 0.

An electron is trapped in a quantum dot, in which it is continued to a very small region in all three dimensions, If the lowest energy transition is to produce a photon of 450nm wavelength, what should be the width of the well (assumed cubic)?

Question: The kinetic energy of hydrogen atom wave functions for which lis its minimum value of 0 is all radial. This is the case for the 1s and 2s states. The 2 p state has some rotational kinetic energy and some radial. Show that for very large n, the states of largest allowed lhave essentially no radial kinetic energy. Exercise 55 notes that the expectation value of the kinetic energy (including both rotational and radial) equals the magnitude of the total energy. Compare this magnitude with the rotational energy alone,L2/2mr2
,assuming that n is large. That lis as large as it can be, and thatr≡n2a0.

A hydrogen atom electron is in a 2p state. If no experiment has been done to establish a z-component of angular momentum, the atom is equally likely to be found with any allowed value of LZ. Show that if the probability densities for these different possible states are added (with equal weighting), the result is independent of both ϕandθ

The only visible spectral lines of hydrogen are four Balmer series lines noted at the beginning of Section 7.3. We wish to cause hydrogen gas to glow with its characteristic visible colors.

(a) To how high an energy level must the electrons be exited?

(b) Energy is absorbed in collisions with other particles. Assume that after absorbing energy in one collision, an electron jumps down through lower levels so rapidly that it is in the ground state before another collision occurs. If an electron is to be raised to the level found in part (a), how much energy must be available in a single collision?

(c) If such energetic collisions are to be affected simply by heating the gas until the average kinetic energy equals the desired upward energy jump, what temperature would be required? (This explains why heating is an impractical way to observe the hydrogen spectrum. Instead, the atoms are ionized by strong electric fields, as is the air when a static electric spark passes through.)
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