• 7.1 Application of the Schrödinger Equation to the Hydrogen Atom
• 7.2 Solution of the Schrödinger Equation for Hydrogen
• 7.3 Quantum Numbers
• 7.4 Magnetic Effects on Atomic Spectra – The Normal Zeeman Effect

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CHAPTER 7�The Hydrogen Atom

This spherical system has very high symmetry causing very high degeneracy of the wavefunctions

Lecture a

Labelling of corresponding video

6.5: Three-Dimensional Infinite-Potential Well V

• The wave function must be a function of all three spatial coordinates. We begin with the conservation of energy
• Multiply this by the wave function to get

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• Now consider momentum as an operator acting on the wave function. In this case, the operator must act twice on each dimension. Given:

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• The three dimensional Schrödinger wave equation is

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Laplace operator

Time independent Schroedinger equation (6.43)

Particle in3-D box

Use 3 quantum numbers n

Problem6.26

Find the energies of the second, third, fourth, and fifth levels for the three dimensional cubical box. Which energy levels are degenerate?

A given state is degenerate when there is more than one wave function for a given energy

For a cubical box L1=L2=L3=L

ground state wavefunction E1 is not degenerate

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Degeneracy

• Analysis of the Schrödinger wave equation in three dimensions introduces three quantum numbers that quantize the energy.

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• A quantum state is degenerate when there is more than one wave function for a given energy.

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• Degeneracy results from particular properties of the potential energy function that describes the system. A perturbation of the potential energy can remove the degeneracy.

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• Use the Schrödinger wave equation for molecules

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• We can remove the degeneracy by applying a magnetic field to the atom or molecule

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Rectangular box

6.6: Simple Harmonic Oscillator

• Simple harmonic oscillators describe many physical situations: springs, diatomic molecules and atomic lattices.

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• Consider the Taylor expansion of a potential function:

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Redefining the minimum potential and the zero potential, we have

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Substituting this into the wave equation:

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Let and which yields .

The pendulum is a simple harmonic oscillator , Foucault pendulum(see miscellaneous on SIBOR )

© 2016 Pearson Education, Inc.

Parabolic Potential Well

• If the lowest energy level is zero, this violates the uncertainty principle.
• The wave function solutions are where H_n(x) are Hermite polynomials of order n.

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• In contrast to the particle in a box, where the oscillatory wave function is a sinusoidal curve, in this case the oscillatory behavior is due to the polynomial, which dominates at small x. The exponential tail is provided by the Gaussian function, which dominates at large x.

Analysis of the Parabolic Potential Well

• The energy levels are given by

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• The zero point energy is called the Heisenberg limit:

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• Classically, the probability of finding the mass is greatest at the ends of motion and smallest at the center (that is, proportional to the amount of time the mass spends at each position).
• Contrary to the classical one, the largest probability for this lowest energy state is for the particle to be at the center.

Hermite polynomial functions are shown above

A hydrogen molecule can be approximated a simple harmonic oscillator with force constant k=1.1x10^3 N/m

Deuteron in a nucleus

3A

• 7.1 Application of the Schrödinger Equation to the Hydrogen Atom
• 7.2 Solution of the Schrödinger Equation for Hydrogen
• 7.3 Quantum Numbers
• 7.4 Magnetic Effects on Atomic Spectra – The Normal Zeeman Effect

​

CHAPTER 7�The Hydrogen Atom

7.1: Application of the Schrödinger Equation to the Hydrogen Atom

• The approximation of the potential energy of the electron-proton system is electrostatic:

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• Rewrite the three-dimensional time-independent Schrödinger Equation.

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For Hydrogen-like atoms (He⁺ or Li⁺⁺)

• Replace e² with Ze² (Z is the atomic number)
• Use appropriate reduced mass μ

Uranium is a chemical element with the symbol U and atomic number Z=92

Application of the Schrödinger Equation

• The potential (central force) V(r) depends on the distance r between the proton and electron.

Transform to spherical polar coordinates because of the radial symmetry.

Insert the Coulomb potential into the transformed Schrödinger equation.

Application of the Schrödinger Equation

• The wave function ψ is a function of r, θ, .

Equation is separable.

Solution may be a product of three functions.

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• We can separate Equation 7.3 into three separate differential equations, each depending on one coordinate: r, θ, or .

Equation 7.3

Divide and conquer !!

7.2: Solution of the Schrödinger Equation for Hydrogen

• Substitute Eq (7.4) into Eq (7.3) and separate the resulting equation into three equations: R(r), f(θ), and g( ).

Separation of Variables

• The derivatives from Eq (7.4)

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• Substitute them into Eq (7.3)

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• Multiply both sides of Eq (7.6) by r² sin² θ / Rfg

Solution of the Schrödinger Equation

• Only r and θ appear on the left side and only appears on the right side of Eq (7.7)
• The left side of the equation cannot change as changes.
• The right side cannot change with either r or θ.

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• Each side needs to be equal to a constant for the equation to be true.

Set the constant −m_ℓ² equal to the right side of Eq (7.7)

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• It is convenient to choose a solution to be .

-------- azimuthal equation

Eq (7.8)

Properties of Valid Wave Functions

Boundary conditions

1. In order to avoid infinite probabilities, the wave function must be finite everywhere.
2. In order to avoid multiple values of the probability, the wave function must be single valued.
3. For finite potentials, the wave function and its derivative must be continuous. This is required because the second-order derivative term in the wave equation must be single valued. (There are exceptions to this rule when V is infinite.)
4. In order to normalize the wave functions, they must approach zero as x approaches infinity.

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Solutions that do not satisfy these properties do not generally correspond to physically realizable circumstances.

Not normalizable

Solution of the Schrödinger Equation

• satisfies Eq (7.8) for any value of m_ℓ.
• The solution be single valued in order to have a valid solution for any , which is

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• m_ℓ to be zero or an integer (positive or negative) for this to be true.
• If Eq (7.8) were positive, the solution would not be realized.

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• Set the left side of Eq (7.7) equal to −m_ℓ² and rearrange it.

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• Everything depends on r on the left side and θ on the right side of the equation.

Solution of the Schrödinger Equation

• Set each side of Eq (7.9) equal to constant ℓ(ℓ + 1).

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• Schrödinger equation has been separated into three ordinary second-order differential equations [Eq (7.8), (7.10), and (7.11)], each containing only one variable.

----Radial equation

----Angular equation

This was divide

Solution of the Radial Equation

• The radial equation is called the associated Laguerre equation and the solutions R that satisfy the appropriate boundary conditions are called associated Laguerre functions.

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• Assume the ground state has ℓ = 0 and this requires m_ℓ = 0.

Eq (7.10) becomes

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• The derivative of yields two terms.

Write those terms and insert Eq (7.1)

Next conquer

Solution of the Radial Equation

• Try a solution

A is a normalization constant.

a₀ is a constant with the dimension of length.

Take derivatives of R and insert them into Eq (7.13).

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• To satisfy Eq (7.14) for any r is for each of the two expressions in parentheses to be zero.

Set the second parentheses equal to zero and solve for a₀.

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Note: ground state has l=0

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Set the first parentheses equal to zero and solve for E.

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Both equal to the Bohr result

Quantum Numbers

• The appropriate boundary conditions to Eq (7.10) and (7.11) leads to the following restrictions on the quantum numbers ℓ and m_ℓ:
• ℓ = 0, 1, 2, 3, . . .
• m_ℓ = −ℓ, −ℓ + 1, . . . , −2, −1, 0, 1, 2, . ℓ . , ℓ − 1, ℓ
• |m_ℓ| ≤ ℓ.

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• The predicted energy level is

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ℓ <n

Hydrogen Atom Radial Wave Functions

• First few radial wave functions R_nℓ

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• Subscripts on R specify the values of n and ℓ

Solution of the Angular and Azimuthal Equations

• The solutions for Eq (7.8) are
• Solutions to the angular and azimuthal equations are linked because both have m_ℓ
• Group these solutions together into functions

---- spherical harmonics

Normalized Spherical Harmonics

Solution of the Angular and Azimuthal Equations

• The radial wave function R and the spherical harmonics Y determine the probability density for the various quantum states. The total wave function depends on n, ℓ, and m_ℓ. The wave function becomes

Problem7.8

The wave function Ψ for the ground state of hydrogen is given by

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Ψ₁₀₀(r,φ,θ) = A e^-r/ao

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Find the constant A that will normalize this wave function over all space.

7.3: Quantum Numbers

The three quantum numbers:

• n Principal quantum number
• ℓ Orbital angular momentum quantum number
• m_ℓ Magnetic quantum number

The boundary conditions:

• n = 1, 2, 3, 4, . . . Integer
• ℓ = 0, 1, 2, 3, . . . , n − 1 Integer
• m_ℓ = −ℓ, −ℓ + 1, . . . , 0, 1, . . . , ℓ − 1, ℓ Integer

The restrictions for quantum numbers:

• n > 0
• ℓ < n
• |m_ℓ| ≤ ℓ

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1) For what levels in the hydrogen atom will we not find l=2 states??

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a) n = 4, 5

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b) n = 3, 4

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c) n = 2, 1

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d) n = 5, 6

Clicker question

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2) Which of the following states of the hydrogen atom is allowed?

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a) n = 6, l = 2, m_l = 0

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b) n = 2, l = 2, m_l = 0

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c) n = 5, l = 2, m_l = 3

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d) n = 1, l = 2, m_l = 1

Clicker question

Problem7.11

List all quantum numbers (n,l,m_l) for the n=5 level in atomic hydrogen.

l<5 Degeneracy : Number of m states namely 2l+1

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total number of states 9+7+5+3+1=25

Principal Quantum Number n

• It results from the solution of R(r) in Eq (7.4) because R(r) includes the potential energy V(r).

The result for this quantized energy is

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• The negative means the energy E indicates that the electron and proton are bound together.

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Lecture b

Orbital Angular Momentum Quantum Number ℓ

• It is associated with the R(r) and f(θ) parts of the wave function.

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• Classically, the orbital angular momentum with L = mv_orbitalr.

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• ℓ is related to L by .
• This curious dependence is a wave phenomenon and will be shown later

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• In an ℓ = 0 state, .

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It disagrees with Bohr’s semi-classical “planetary” model of electrons orbiting a nucleus L = nħ.

Orbital Angular Momentum Quantum Number ℓ

• A certain energy level is degenerate with respect to ℓ when the energy is independent of ℓ.

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• Use letter names for the various ℓ values
• ℓ = 0 1 2 3 4 5 . . .
• Letter = s p d f g h . . .

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• Atomic states are referred to by their n and ℓ
• A state with n = 2 and ℓ = 1 is called a 2p state
• The boundary conditions require n > ℓ

Degeneracy : Number of m states namely 2l+1

The Uncertainty principle� forbids this classical fig 8.6 and QM requires “Fuzzyness” of the angular momentum

Space Quantization-Magnetic Quantum Number m_ℓ

• The angle is a measure of the rotation about the z axis.
• The solution for specifies that m_ℓ is an integer and related to the z component of L.

• The relationship of L, L_z, ℓ, and m_ℓ for ℓ = 2.
• is fixed.
• Because L_z is quantized, only certain orientations of are possible and this is called space quantization.

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Magnetic Quantum Number m_ℓ

• Quantum mechanics allows to be quantized along only one direction in space. Because of the relation L² = L_x² + L_y² + L_z² the knowledge of a second component would imply a knowledge of the third component because we know .

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• We expect the average of the angular momentum components squared to be

Use a math table for the summation result

Since the sum

Fuzzyness of angular momentum

Honda 600RR

Who races this bike?

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Why can anybody race it, if he just

dares to go fast?

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The oval track of the Texas

World Speedway allowed speeds

of 250 mph.

Texas World Speedway is a defunct American racetrack which was built in 1969 and was one of only eight superspeedways of two miles or greater in the United States used for racing, On September 18, 2017 an article on Jalopnik was posted confirming the closure of Texas World Speedway which was being used as a dumping ground for vehicles flooded out by Hurricane Harvey. TWS is the first superspeedway to shut down. The entire 600-acre facility was being leased to Copart Inc.

7.4: Magnetic Effects on Atomic Spectra—The Normal Zeeman Effect

• The Dutch physicist Pieter Zeeman showed the spectral lines emitted by atoms in a magnetic field split into multiple energy levels. It is called the Zeeman effect.

External magnetic field reduces spherical symmetry

Normal Zeeman effect:

• A spectral line is split into three lines.
• Consider the atom to behave like a small magnet.
• The current loop has a magnetic moment μ = IA and the period T = 2πr / v.
• Think of an electron as an orbiting circular current loop of I = dq / dt around the nucleus.
• where L = mvr is the magnitude of the orbital

angular momentum

The Normal Zeeman Effect

• The angular momentum is aligned with the magnetic moment, and the torque between and causes a precession of .

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Where μ_B = eħ / 2m is called a Bohr magneton.

• cannot align exactly in the z direction and �has only certain allowed quantized orientations.

• Since there is no magnetic field to align them, point in random directions. The dipole has a potential energy

Precession frequency

The Normal Zeeman Effect

• The potential energy is quantized due to the magnetic quantum number m_ℓ.

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• When a magnetic field is applied, the 2p level of atomic hydrogen is split into three different energy states with energy difference of ΔE = μ_BB Δm_ℓ.

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Spectrum of atomic hydrogen

The Normal Zeeman Effect

• A transition from 2p to 1s

History of Hydrogen Spectroscopy

Theodor Hänsch 2012 Nobel Laureate

Space quantizationin the Stern Gerlach experiment

Stern Gerlach used silver atoms

The Normal Zeeman Effect

• An atomic beam of particles in the ℓ = 1 state pass through a inhomogeneous magnetic field along the z direction.

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• The m_ℓ = +1 state will be deflected down, the m_ℓ = −1 state up, and the m_ℓ = 0 state will be undeflected.
• If the space quantization were due to the magnetic quantum number m_ℓ, m_ℓ states is always odd (2ℓ + 1) and should have produced an odd number of lines.

7.5: Intrinsic Spin

• Samuel Goudsmit and George Uhlenbeck in Holland proposed that the electron must have an intrinsic angular momentum and therefore a magnetic moment.

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• Paul Ehrenfest showed that the surface of the spinning electron should be moving faster than the speed of light!

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• In order to explain experimental data, Goudsmit and Uhlenbeck proposed that the electron must have an intrinsic spin quantum number s = ½.

Intrinsic Spin

• The spinning electron reacts similarly to the orbiting electron in a magnetic field.
• We should try to find L, L_z, ℓ, and m_ℓ.
• The magnetic spin quantum number m_s has only two values, m_s = ±½.

The electron’s spin will be either “up” or “down” and can never be spinning with its magnetic moment μ_s exactly along the z axis.

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The intrinsic spin angular momentum vector .

How fast does the earth rotate?

Approximately 1,675 km/h (1,040 mph) at the equator, slightly less near the poles.

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Clicker - Questions

8. Stern and Gerlach performed an experiment that showed the space quantization of silver atoms in an inhomogeneous magnetic field. Their experiment demonstrated that

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a. space quantization is a property that only exists for energy levels, governed by quantum number n.

b. the differences in magnetic moment of the atom demonstrated space quantization in external magnetic fields.

c. the classically defined Bohr magneton was inaccurate because it did not take into account the space quantization of external magnetic fields within the atom.

d. an additional angular momentum factor within the atom was causing the observed space quantization.

• 1. For an atom of hydrogen where the quantum number n = 3, how many possible independent configurations of the atom exist (consider principal quantum number n, angular momentum quantum number l, magnetic quantum number ml, and magnetic spin quantum number ms)?
• 18
• 8
• 32
• 26
• infinite

Clicker - Questions

Lecture c

Intrinsic Spin

• The magnetic moment is .
• The coefficient of is −2μ_B as with is a consequence of theory of relativity.

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• The gyromagnetic ratio (ℓ or s).
• g_ℓ = 1 and g_s = 2, then

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• The z component of .
• In ℓ = 0 state

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• Apply m_ℓ and the potential energy becomes

no splitting due to .

there is space quantization due to the intrinsic spin.

and

The numerical factor relating the magnetic moment to each angular momentum vector is the gyromagnetic ratio

Geonium

Hans Georg Dehmelt (9 September 1922 – 7 March 2017)was a German and American physicist, who was awarded a Nobel Prize in Physics in 1989, for co-developing the ion trap technique (Penning trap) with Wolfgang Paul,

Results: Endcap Trap

Large ²⁴Mg⁺ - ²⁶Mg⁺ ion crystal (N~10⁴)

Space quantization of the electron spin angular momentum

In the frame of the electron there is an internal magnetic field created by the orbiting proton= doubled splitting

Doublet splitting due to the electron spin magnetic moment

• The gyromagnetic ratio

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• a. is 1 for the magnetic moment associated with the spin and 2 for the magnetic moment associated with the angular momentum.
• b. relates the Bohr magneton to the elementary charge.
• c. does not help explain the result of the Stern and Gerlach experiment.
• d. relates the magnetic moments of spin and angular momentum to the total angular momentum.
• e.gives the values of intrinsic spin quantum number of the electron as 1/2 and -1/2

Clicker - Questions

Problem7.29

Use all four quantum numbers (n,l.m_l,m_s) to write down all possible sets of quantum numbers for the 4f state of atomic hydrogen. What is the total degeneracy?

Problem7.32

Use all four quantum numbers (n,l.m_l,m_s) to write down all possible sets of quantum numbers for the 5d state of atomic hydrogen. What is the total degeneracy?

7.6: Energy Levels and Electron Probabilities

• For hydrogen, the energy level depends on the principle quantum number n.

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• In ground state an atom cannot emit radiation. It can absorb electromagnetic radiation, or gain energy through inelastic bombardment by particles.

Forbidden transitions: 3P-2P, 3d-2S,4F-3S, etc

Selection Rules

• We can use the wave functions to calculate transition probabilities for the electron to change from one state to another.

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Allowed transitions:

• Electrons absorbing or emitting photons to change states when Δℓ = ±1.

Forbidden transitions:

• Other transitions possible but occur with much smaller probabilities when Δℓ ≠ ±1.

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Conservation of angular momentum: photon carries one unit of angular momentum. The atom changes by one unit of angular momentum in the radiation process(the sum =0)

3-D Probability Distribution Functions

• We must use wave functions to calculate the probability distributions of the electrons.

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• The “position” of the electron is spread over space and is not well defined.

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• We may use the radial wave function R(r) to calculate radial probability distributions of the electron.
• The probability of finding the electron in a differential volume element dτ is .

3-D Probability Distribution Functions

• The differential volume element in spherical polar coordinates is

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Therefore,

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• We are only interested in the radial dependence.

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• The radial probability density is P(r) = r²|R(r)|² and it depends only on n and l.

Probability distributions in 3D space( as shown before)

Normalizing a hydrogenic wave function

Radial Probability Distribution Functions

• R(r) and P(r) for the lowest-lying states of the hydrogen atom

n=1

n=2

n=3

Horizontal axis in units of the Bohr radius

Lecture d

3-D Probability Distribution Functions

• The probability density for the hydrogen atom for three different electron states

The James Webb Space Telescope is a joint NASA–ESA–CSA space telescope that is planned to succeed the Hubble Space Telescope as NASA's flagship astrophysics mission. Wikipedia

Launch date: December 18, 2021

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The Hubble Space Telescope makes one orbit around Earth every 95 minutes.

The deep-space observatory observed an exoplanet with evidence of silicate-rich clouds; the brown dwarf is nearly 20 times the size of Jupiter. The brown dwarf is called VHS 1256 b and orbits two small red dwarf stars, 72 light-years from Earth. September02,2022

NASA’s Space Launch System (SLS) rocket with the Orion spacecraft aboard is seen atop the mobile launcher at Launch 39B at NASA’s Kennedy Space Center in Florida. Artemis I mission is the first integrated test of the agency’s deep space exploration systems: the Space Launch System rocket, Orion spacecraft, and supporting ground systems. The mission is the first in a series of increasingly complex missions to the Moon. With Artemis missions, NASA will land the first woman and first person of color on the Moon,

Space Station

The SPHERES Tether Slosh investigation combines fluid dynamics equipment with robotic capabilities aboard the station. In space, the fuels used by spacecraft can slosh around in unpredictable ways making space maneuvers difficult. This investigation uses two SPHERES robots tethered to a fluid-filled container covered in sensors to test strategies for safely steering spacecraft such as dead satellites that might still have fuel in the tank.

Selfie of the Mars Rover in a Dust Storm

Lecture e

Find whether the following transitions are allowed, and if they are, find the energy involved and whether the photon is absorbed or emitted for the hydrogen atom:

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(a)(5, 2, 1, 1/2) (5, 2, 2,1/2)

(b)(4, 3, 0, 1/2) (4, 2, 1, -1/2)

(c)(5, 2, -2, -1/2) (1, 0, 0, -1/2)

(d)(2, 1, 1, 1/2) (4, 2, 1, 1/2)

4. (a)

l = 0

is forbidden

(b) allowed but with

 n = 0

there is no energy difference unless an external magnetic

field is present

(c)

 = −2

is forbidden

(d) allowed with absorbed photon of energy

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E= E0(1/2^2-1/4^2)=2.55 eV

E0= ground state of hydrogen

What is the probability that an electron in the 3dstate is located at a radius greater than a0?

Radial Probability Distribution Functions

• R(r) and P(r) for the lowest-lying states of the hydrogen atom

n=1

n=2

n=3

Horizontal axis in units of the Bohr radius

Consider a hydrogen-like atom such as He+ or Li++ that has a single electron outside a nucleus of charge 1Ze. (a) Rewrite the Schrödinger equation with the new Coulomb potential. (b) What change does this new potential have on the separation of variables? (c) Will the radial wave functions be affected? Ex-plain. (d) Will the spherical harmonics be affected? Explain.

Radial equation

(d) no, there is no angular dependence

Problem 7.31

The 21-cm line transition of atomic hydrogen results from a spin-flip transition for the electron in the parallel state of the n=1 state. What temperature in interstellar space gives a hydrogen atom enough energy (5.9x10^-6eV) to excite another hydrogen atom in a collision?

Integrate by substitution u=2x

du/dx=2 dx=du/2