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Chapter 7: Q1CQ (page 278)

What is a quantum number, and how does it arise?

Short Answer

Expert verified

A quantum number is a number that portrays the condition of a molecule. A bunch of quantum numbers might be expected to portray the state completely.Quantum numbers come up when we tackle the Schrodinger condition with forced limit conditions to get a genuinely satisfactory arrangement.

Step by step solution

01

Step 1:Define quantum numbers

In science and quantum physical science, quantum numbers exhibit the upsides of preserved amounts in the elements of a quantum framework. Quantum numbers are used to show the state of an atom or molecule. It gives us all the necessary information that is important to understand the nature, properties, and characteristics of the entity.

02

The rise of quantum numbers

A quantum number is a number that portrays the condition of a molecule. A bunch of quantum numbers might be expected to portray the state completely. Quantum numbers come up when we tackle the Schrodinger condition with forced limit conditions to get a genuinely satisfactory arrangement. These limit conditions led to standing waves and quantized detectable amounts (like energy) that are portrayed by quantum numbers.

The four quantum numbers are Principal quantum number, azimuthal quantum number, magnetic quantum number, and Spin quantum number. Each quantum number tells a different story about the respective atom/ molecule.The principal quantum number gives information about the shells, azimuthal quantum numbers tell us about the subshells, and magnetic and spin quantum numbers tell us about how electrons are settled in the orbits.

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Most popular questions from this chapter

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.

We have noted that for a given energy, as lincreases, the motion is more like a circle at a constant radius, with the rotational energy increasing as the radial energy correspondingly decreases. But is the radial kinetic energy 0 for the largest lvalues? Calculate the ratio of expectation values, radial energy to rotational energy, for the(n,l,mt)=(2.1,+1)state. Use the operators

KE铃�rad=-h22m1r2鈭�鈭�r(r鈭�鈭�r)KE铃�rad=h2l(l+1)2mr2

Which we deduce from equation (7-30).

In general, we might say that the wavelengths allowed a bound particle are those of a typical standing wave,位=2L/n , where is the length of its home. Given that 位=h/p, we would have p=nh/2L, and the kinetic energy, p2/2m, would thus be n2h2/8mL2. These are actually the correct infinite well energies, for the argumentis perfectly valid when the potential energy is 0 (inside the well) and is strictly constant. But it is a pretty good guide to how the energies should go in other cases. The length allowed the wave should be roughly the region classically allowed to the particle, which depends on the 鈥渉eight鈥� of the total energy E relative to the potential energy (cf. Figure 4). The 鈥渨all鈥� is the classical turning point, where there is nokinetic energy left: E=U. Treating it as essentially a one-dimensional (radial) problem, apply these arguments to the hydrogen atom potential energy (10). Find the location of the classical turning point in terms of E , use twice this distance for (the electron can be on both on sides of the origin), and from this obtain an expression for the expected average kinetic energies in terms of E . For the average potential, use its value at half the distance from the origin to the turning point, again in terms of . Then write out the expected average total energy and solve for E . What do you obtain

for the quantized energies?

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

Residents of flatworld-a two-dimensional world far, far away-have it easy. Although quantum mechanics of course applies in their world, the equations they must solve to understand atomic energy levels involve only two dimensions. In particular, the Schrodinger equation for the one-electron flatrogen atom is

-h2m1r鈭�鈭�r(r鈭�鈭�r)蠄(r,胃)-h2m1r2鈭�2鈭�胃2蠄(r,胃)+U(r)蠄(r,胃)=E蠄(r,胃)

(a) Separate variables by trying a solution of the form 蠄(r,胃)=R(r)鈯�(胃), then dividing byR(r)鈯�(胃) . Show that the 胃equation can be written

d2d胃2鈯�(胃)=C鈯�(胃)

Here,(C) is the separation constant.

(b) To be physically acceptable,鈯�(胃) must be continuous, which, since it involves rotation about an axis, means that it must be periodic. What must be the sign of C ?

(c) Show that a complex exponential is an acceptable solution for鈯�(胃) .

(d) Imposing the periodicity condition find allowed values ofC .

(e) What property is quantized according of C .

(f) Obtain the radial equation.

(g) Given thatU(r)=-b/r , show that a function of the formR(r)=er/a is a solution but only if C certain one of it, allowed values.

(h) Determine the value of a , and thus find the ground-state energy and wave function of flatrogen atom.
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