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Chapter 4: Q14P (page 156)

What is the most probable value of r, in the ground state of hydrogen? (The answer is not zero!) Hint: First you must figure out the probability that the electron would be found between r and r + dr.

Short Answer

Expert verified

The most probable value of r is in the ground state of hydrogen.

Step by step solution

01

Expression of probability

The expression for the probability is given as follows,

P=|ψ|24πr2dr=4a3e-2r/ar2dr=p(r)dr

Here, the value of is , 4a3r2e-2r/aand the wave function in the ground state of hydrogen,ψ=1πa3e-r/a.

02

Determination of the most probable value of r

Take the derivative of p(r)=4a3r2e-2r/awith respect to r .

role="math" localid="1657772519231" dprdr=ddr4a3r2e-2r/a=4a32re-2r/a+r22ae-2r/a=8ra3e-2r/a1-ra

Equate the obtained value to zero.

dprdr=08ra3e-2r/a1-ra=0

For the value of r equate role="math" localid="1657772026855" 1-rato zero.

1-ra=0r=a

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Most popular questions from this chapter

(a) If you measured the component of spin angular momentum along the x direction, at time t, what is the probability that you would get +h/2?

(b) Same question, but for the ycomponent.

(c) Same, for the z component.

Consider the observablesA=x2andB=Lz .

(a) Construct the uncertainty principle forσAσB

(b) EvaluateσB in the hydrogen stateψn/m .

(c) What can you conclude about<xy>in this state?

Deduce the condition for minimum uncertainty inSx andSy(that is, equality in the expression role="math" localid="1658378301742" σSxσSy≥(ħ/2)|<Sz>|, for a particle of spin 1/2 in the generic state (Equation 4.139). Answer: With no loss of generality we can pick to be real; then the condition for minimum uncertainty is that bis either pure real or else pure imaginary.

a) Check that Arj1(kr)satisfies the radial equation with V(r)=0and I=1.

(b) Determine graphically the allowed energies for the infinite spherical well, when I=1. Show that for large n,En1≈(h2π2/2ma2)(n+1/2)2. Hint: First show that j1(x)=0⇒x=tanx. Plot xandtanxon the same graph, and locate the points of intersection.

An electron is at rest in an oscillating magnetic field

B=B0cos(ωt)k^

whereB0 andω are constants.

(a) Construct the Hamiltonian matrix for this system.

(b) The electron starts out (at t=0 ) in the spin-up state with respect to the x-axis (that is:χ(0)=χ+(x)). Determine X(t)at any subsequent time. Beware: This is a time-dependent Hamiltonian, so you cannot get in the usual way from stationary states. Fortunately, in this case you can solve the timedependent Schrödinger equation (Equation 4.162) directly.

(c) Find the probability of getting-h/2 , if you measure Sx. Answer:

sin2(γB02ωsin(ωt))

(d) What is the minimum field(B0) required to force a complete flip inSx ?
See all solutions

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