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Chapter 7: Q73E (page 283)

Verify for the angular solutions 鈯�(胃)蠁(蠒)of Table 7.3 that replacing 蠒 with 蠒+蟺 and replacing 胃 with 蟺-胃gives 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,蠁(蠒)=eiml蠒,

Where, 蠒is the colatitude and m1is the magnetic quantum number.

By replacing 蠒with(蠒+蟺) , you get,
role="math" localid="1659699420914" 蠁(蠒+蟺)=eiml(蠒+蟺)=eiml(蟺)=eiml蠒(cosml蟺+isinml蟺)

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,蠁(蠒) the changes sign when is odd and remains changed otherwise.

02

Replacing θ with π-θ :

In the function 鈯�(胃), here, 胃is the colatitude

By replacing 胃with蟺-胃 you get,
cos(蟺-胃)=-cos(胃)sin(蟺-胃)=sin(胃)

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

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Most popular questions from this chapter

Calculate the electric dipole moment p and estimate the transition time for a hydrogen atom electron making an electric dipole transition from the to the (n,l,m)=(2,1,+1) ground state. Comment on the relationship of the result to that in Example 7.11.

Calculate the 鈥渟eries limit鈥� of the Lyman series of spectral lines. This is defined as the shortest wavelength possible of a photon emitted in a transition from a higher initial energy level to the ni=1 final level. (Note: In figure 7.5, the spectral lines of the series 鈥渃rowd together鈥� at the short-wavelength end of the series).

Consider a vibrating molecule that behaves as a simple harmonic oscillator of mass 10-27kg, spring constant 103N/m and charge is +e , (a) Estimate the transition time from the first excited state to the ground state, assuming that it decays by electric dipole radiation. (b) What is the wavelength of the photon emitted?

A comet of 1014鈥�kg mass describes a very elliptical orbit about a star of mass3脳1030鈥�kg , with its minimum orbit radius, known as perihelion, being role="math" localid="1660116418480" 1011m and its maximum, or aphelion, 100 times as far. When at these minimum and maximum

radii, its radius is, of course, not changing, so its radial kinetic energy is 0, and its kinetic energy is entirely rotational. From classical mechanics, rotational energy is given by L22I, where Iis the moment of inertia, which for a 鈥減oint comet鈥� is simply mr2.

(a) The comet鈥檚 speed at perihelion is6.2945脳104鈥�m/s . Calculate its angular momentum.

(b) Verify that the sum of the gravitational potential energy and rotational energy are equal at perihelion and aphelion. (Remember: Angular momentum is conserved.)

(c) Calculate the sum of the gravitational potential energy and rotational energy when the orbit radius is 50 times perihelion. How do you reconcile your answer with energy conservation?

(d) If the comet had the same total energy but described a circular orbit, at what radius would it orbit, and how would its angular momentum compare with the value of part (a)?

(e) Relate your observations to the division of kinetic energy in hydrogen electron orbits of the same nbut different I.

The space between two parallel metal plates is filled with an element in a gaseous stale. Electrons leave one plate at negligible speed and are accelerated toward the other by a potential difference鈭�Vapplied between the plates. As 鈭�Vis increased from 0, the electron current increases more or less linearly, but when 鈭�Vreaches 4.9 V , the current drops precipitously. From nearly 0 , it builds again roughly linearly as 鈭�Vis increased beyond 4.9 V .

(a) How can the presence of the gas explain these observations?

(b) The Gas emits pure 鈥渓ight鈥� when 鈭�Vexceeds 4.9 V . What is its wavelength?
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