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Q. 36

Physics For Scientists & Engineers
Found in: Page 1296

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Short Answer

II In general, an atom can have both orbital angular momentum and spin angular momentum. The total angular momentum is defined to be J=L+S. The total angular momentum is quantized in the same way as L and S. That is, J=jj+1h, where j is the total angular momentum quantum number. The z- component of J is Jz=Lz+Sz=mjh, where mj goes in integer steps from -j to +j. Consider a hydrogen atom in a p state, with l=1.

a. role="math" localid="1648549390842" Lz has three possible values and Sz has two. List all possible combinations of Lz and Sz. For each, compute Jz and determine the quantum number mj. Put your results in a table.

b. The number of values of Jz that you found in part a is too many to go with a single value of j. But you should be able to divide the values of Jz into two groups that correspond to two values of j. What are the allowed values of j? Explain. In a classical atom, there would be no restrictions on how the two angular momenta L and S can combine. Quantum mechanics is different. You've now shown that there are only two allowed ways to add these two angular momenta.

According to the wave mechanics, the orbital quantum number / is restricted to the integer values from 0 to n-1. That is, all the possible values of the orbital quantum number / depend upon the principal quantum number n. They are expressed as


(a) 32 and 12

(b) Values of j are 12,32

See the step by step solution

Step by Step Solution


The magnetic quantum number m is restricted to the integer values from -/ to +/. That is, all the possible values of the magnetic quantum number m depend upon the orbital quantum number l. Mathematically, it is expressed as


The spin quantum number ms correspond to different orientations of the axis of the spin of the electron. There are basically two values of the spin quantum number ms, namely -12 and +12.

Magnitude of the spin angular quantum number :

The magnitude of the spin angular quantum number obeys the quantization rule. Using the quantization condition, the equation for the magnitude of the spin angular momentum Ls is given as,


Here, h is Planck's constant and is equal to h divided by 2p.

Part (a) Step 1:

Forl=1, the magnetic quantum number varies as,


By using the equation (1), the spin angular momentum varies as,


For each values of m, there are two s states as,

+12 and -12

There are S2=12h and -12h for each L2.

The total angular momentum quantum number is,

Js=L2+S2and Jz=mjh.

Now for L2=h we have two values of Js which are,

h+12h=32h and h-12h=12h

Then the corresponding mj values are

32 and 12.

Proceeding in the same way, we get the following table:

" width="9" height="19" role="math" style="max-width: none; vertical-align: -4px;" localid="1648552354339">-h12h-12h-12

Part (b) step 1:

As j=l+m, and we have l=1 and m=+12 or -12

The allowed values of j are

1+12=32 and 1-12=12

Allowed values of j are 12,32.

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