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Chapter 28: Problem 36

How many electron states of the H atom have the quantum numbers \(n=3\) and \(\ell=1 ?\) Identify each state by listing its quantum numbers.

Short Answer

Expert verified
Answer: There are 3 electron states with the quantum numbers \(n=3\) and \(\ell=1\) for hydrogen atom. These states are: \((3, 1, -1)\), \((3, 1, 0)\), and \((3, 1, 1)\).

Step by step solution

01

Find the possible values of the magnetic quantum number \(m_\ell\)

Given that \(\ell = 1\), we can find the possible values of the magnetic quantum number \(m_\ell\) using the range \((-\ell, -\ell+1, ..., 0, ..., \ell-1, \ell)\). So, for \(\ell = 1\), the possible values of \(m_\ell\) will be \((-1, 0, 1)\).
02

Count the number of electron states and identify them

Now that we know the possible values of \(m_\ell\), we can count the number of electron states and identify each state by listing its quantum numbers \((n, \ell, m_\ell)\). There are in total 3 values of \(m_\ell = -1, 0, 1\). The electron states with quantum numbers \((n, \ell, m_\ell)\) are as follows: 1. State 1: \((3, 1, -1)\) 2. State 2: \((3, 1, 0)\) 3. State 3: \((3, 1, 1)\) There are 3 electron states of the hydrogen atom with the given quantum numbers \(n = 3\) and \(\ell = 1\).

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Most popular questions from this chapter

(a) Show that the ground-state energy of the hydrogen atom can be written \(E_{1}=-k e^{2} /\left(2 a_{0}\right),\) where \(a_{0}\) is the Bohr radius. (b) Explain why, according to classical physics, an electron with energy \(E_{1}\) could never be found at a distance greater than \(2 a_{0}\) from the nucleus.
A hydrogen atom has a radius of about \(0.05 \mathrm{nm}\) (a) Estimate the uncertainty in any component of the momentum of an electron confined to a region of this size. (b) From your answer to (a), estimate the electron's kinetic energy. (c) Does the estimate have the correct order of magnitude? (The ground-state kinetic energy predicted by the Bohr model is \(13.6 \mathrm{eV} .\) )
The energy-time uncertainty principle allows for the creation of virtual particles, that appear from a vacuum for a very brief period of time $\Delta t,\( then disappear again. This can happen as long as \)\Delta E \Delta t=\hbar / 2,\( where \)\Delta E$ is the rest energy of the particle. (a) How long could an electron created from the vacuum exist according to the uncertainty principle? (b) How long could a shot put with a mass of \(7 \mathrm{kg}\) created from the vacuum exist according to the uncertainty principle?
(a) Show that the number of electron states in a subshell is \(4 \ell+2 .\) (b) By summing the number of states in each of the subshells, show that the number of states in a shell is \(2 n^{2} .\) [Hint: The sum of the first \(n\) odd integers, from 1 to \(2 n-1,\) is \(n^{2} .\) That comes from regrouping the sum in pairs, starting by adding the largest to the smallest: \(1+3+5+\dots+(2 n-5)+(2 n-3)+(2 n-1)\) \(=[1+(2 n-1)]+[3+(2 n-3)]+[5+(2 n-5)]+\cdots\) \(=2 n+2 n+2 n+\cdots=2 n \times \frac{n}{2}=n^{2}\)
A proton and a deuteron (which has the same charge as the proton but 2.0 times the mass) are incident on a barrier of thickness \(10.0 \mathrm{fm}\) and "height" \(10.0 \mathrm{MeV} .\) Each particle has a kinetic energy of $3.0 \mathrm{MeV} .$ (a) Which particle has the higher probability of tunneling through the barrier? (b) Find the ratio of the tunneling probabilities.
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