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Chapter 4: Q16P (page 158)

A hydrogenic atom consists of a single electron orbiting a nucleus with Z protons. (Z=1 would be hydrogen itself,Z=2is ionized helium ,Z=3is doubly ionized lithium, and so on.) Determine the Bohr energies En(Z), the binding energyE1(Z), the Bohr radiusa(Z), and the Rydberg constant R(Z)for a hydrogenic atom. (Express your answers as appropriate multiples of the hydrogen values.) Where in the electromagnetic spectrum would the Lyman series fall, for Z=2and Z=3? Hint: There鈥檚 nothing much to calculate here鈥� in the potential (Equation 4.52) Ze2, so all you have to do is make the same substitution in all the final results.

V(r)=-e24蟺o0藱1r (4.52).

Short Answer

Expert verified

The electronic spectrum would be the Lyman seires fall, forz=2 andz=3 is at,z=2 is3.04脳10-8m.

At z=3is1.35脳10-8m.

Step by step solution

01

Given

The potential is given by:

V(r)=-e24蟺o0藱1r

02

Electronic spectrum

Consider a hydrogen, if irradiated with light it will be excited to another energy level. The energies of these photons are easily calculated from the Bohr energy formula:

En=-1n2me42h2(4韦韦鈭�0)2

Lyman lines range fromni=2toni=鈭�(withnf=1);

the wavelengths range from

03

Finding the electronic spectrum

1位2=R1-14=34R位2=43R

Down to

role="math" localid="1658210586427" 1位1=R1-1鈭�=R鈬�R1=1R

ForZ=2

位1=14R=14(1.097脳107)=2.28脳10-8m

to

位2=13R=3.04脳10-8m,

Ultraviolet.

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Most popular questions from this chapter

(a) Find the eigenvalues and eigenspinors of Sy .

(b) If you measured Syon a particle in the general state X(Equation 4.139), what values might you get, and what is the probability of each? Check that the probabilities add up to 1 . Note: a and b need not be real!

(c) If you measuredSy2 , what values might you get, and with what probabilities?

Use separation of variables in Cartesian coordinates to solve infinite cubical well

V(x,y,z)=0if x,y,z are all between 0 to a;

V(x,y,z)=鈭�Otherwise

a) Find the stationary states and the corresponding energies

b) Call the distinct energies E1,E2,E3,..in the order of increasing energy. Findlocalid="1658127758806" E1,E2,E3,E4,E5,E6determine their degeneracies (that is, the number of different states that share the same energy). Comment: In one dimension degenerate bound states do not occur but in three dimensions they are very common.

c) What is the degeneracy of E14 and why is this case interesting?

(a) Construct the wave function for hydrogen in the state n=4,I=3,m=3. Express your answer as a function of the spherical coordinates r,胃and蠒.

(b) Find the expectation value of role="math" localid="1658391074946" rin this state. (As always, look up any nontrivial integrals.)

(c) If you could somehow measure the observable Lx2+Ly2on an atom in this state, what value (or values) could you get, and what is the probability of each?

A hydrogen atom starts out in the following linear combination of the stationary states n=2, l=1, m=1 and n=2, l=1, m=-1.

蠄(r,0)=12(蠄211+蠄21-1)

(a) Construct蠄(r,t)Simplify it as much as you can.

(b) Find the expectation value of the potential energy,<V>. (Does it depend on t?) Give both the formula and the actual number, in electron volts.

Use equations 4.27 4.28 and 4.32 to construct Y00,Y21Check that they are normalized and orthogonal
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