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Chapter 7: Q15CQ (page 279)

(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?

Short Answer

Expert verified
  1. For a one dimensional particle in a box, the n represents the shell to which electron belongs. The value of n may be 1,2,3,.............n.
  2. In hydrogen, principal quantum number holds a great significance. It can specify the radial distributions of the wave function and the energy values of the atom with different n values
  3. Angular quantum number, l : represents the shape of the orbital of hydrogen Magnetic quantum number, ml: represents the energy levels in sub shell.

Step by step solution

01

Quantum numbers

Quantum numbers are the values which determine the position and energy of an electron in an atom, there are four types of quantum numbers, principal (n) azimuthal (l), magnetic (ml) and spin quantum number (ms).

02

(a) The significance of n for 1D particle in a box

The Quantum-mechanical model of an atom provides us with a few quantum numbers which helps to determine the exact location of electron in an atom.

The first quantum number is the principal quantum number which is represented by n , it represents the shell number to which an electron belongs.

Hence,for a one dimensional particle in a box, the n represents the shell to which electron belongs. The value of n may be1,2,3,.............n

03

(b) To determine the meaning of  for a hydrogen atom

In hydrogen atom, principal quantum number holds a great significance. It can specify the radial distributions of the wave function and the energy values of the atom with different n values.

It also specifies the radial dependence of the radial wave function.

04

(c) To determine the meaning andfor hydrogen atom

Just like principal quantum number, there are three more quantum numbers, out of which two are - angular quantum number, l and the magnetic quantum number, ml.

Hence, Angular quantum number, l : represents the shape of the orbital of hydrogen Magnetic quantum number,ml : represents the energy levels in sub shell.

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Most popular questions from this chapter

For the more circular orbits, 鈩�=n-1and

P(r)鈭�r2ne-2r/na0

a) Show that the coefficient that normalizes this probability is

localid="1660047077408" (2na0)2n+11(2n)!

b) Show that the expectation value of the radius is given by

r炉=n(n+12)a0

and the uncertainty by

螖r=na0n2+14

c) What happens to the ratio螖r/r炉in the limit of large n? Is this large-n limit what would be expected classically?

Question: Consider a cubic 3D infinite well of side length of L. There are 15 identical particles of mass m in the well, but for whatever reason, no more than two particles can have the same wave function. (a) What is the lowest possible total energy? (b) In this minimum total energy state, at what point(s) would the highest energy particle most likely be found? (Knowing no more than its energy, the highest energy particle might be in any of multiple wave functions open to it and with equal probability.)

An electron in a hydrogen atom is in the (n,l,ml) = (2,1,0) state.

(a) Calculate the probability that it would be found within 60 degrees of z-axis, irrespective of radius.

(b) Calculate the probability that it would be found between r = 2a0 and r = 6a0, irrespective of angle.

(c) What is the probability that it would be found within 60 degrees of the z-axis and between r = 2a0 and r = 6a0?

What are the dimensions of the spherical harmonics 螛l,ml(胃)桅ml(蠒)given in Table 7.3? What are the dimensions of theRn,l(r)given in Table 7.4, and why? What are the dimensions ofP(r), and why?

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?
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