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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

Classically, it was expected that an orbiting electron would emit radiation of the same frequency as its orbit frequency. We have often noted that classical behaviour is observed in the limit of large quantum numbers. Does it work in this case? (a) Show that the photon energy for the smallest possible energy jump at the “low-n-end” of the hydrogen energies is 3|E0|/n3, while that for the smallest jump at the “high-n-end” is 2|E0|/n3, where E0is hydrogen’s ground-state energy. (b) Use F=ma to show that the angular velocity of a classical point charge held in orbit about a fixed-point charge by the coulomb force is given by ω=e2/4πε0mr3. (c) Given that r=n2a0, is this angular frequency equal to the minimum jump photon frequency at either end of hydrogen’s allowed energies?

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)∝r2ne-2r/nao

Show that the most probable radius is given by

rmostprobable=n2ao

Here we Pursue the more rigorous approach to the claim that the property quantized according to ml is Lz,

(a) Starting with a straightforward application of the chain rule,

∂∂φ = ∂x∂φ/∂∂x+∂y∂φ∂∂y+∂z∂φ∂∂z

Use the transformations given in Table 7.2 to show that

∂∂φ = -y∂∂x+x∂∂y

(b) Recall that L = r x p. From the z-component of this famous formula and the definition of operators for px and py, argue that the operator for Lz is -ih∂∂φ..

(c) What now allows us to say that our azimuthal solutioneimlφ has a well-defined z-component of angular momentum and that is value mlh.

Explicitly verify that the simple function Rr=Aebrcan be made to satisfy radial equation (7-31), and in so doing, demonstrate what its angular momentum and energy must be.

When applying quantum mechanics, we often concentrate on states that qualify as “orthonormal”, The main point is this. If we evaluate a probability integral over all space of ϕ1*ϕ1or of ϕ2*ϕ2, we get 1 (unsurprisingly), but if we evaluate such an integral forϕ1*ϕ2orϕ2*ϕ1 we get 0. This happens to be true for all systems where we have tabulated or actually derived sets of wave functions (e.g., the particle in a box, the harmonic oscillator, and the hydrogen atom). By integrating overall space, show that expression (7-44) is not normalized unless a factor of 1/2is included with the probability.

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