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Page 1: Atom hidrogen

Schrödinger Equation

Angular Momentum

Solution of the Schrödinger Equation

for Hydrogen

The Hydrogen Atom

The atom of modern physics can be symbolized only through a partial differential

equation in an abstract space of many dimensions. All its qualities are inferential; no

material properties can be directly attributed to it. An understanding of the atomic world

in that primary sensuous fashion…is impossible.

- Werner Heisenberg

Werner Heisenberg

(1901-1976)

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1. Schrödinger Equation

The time-independent Schrodinger equation for atom hydrogen:

Use the three-dimensional time-independent Schrödinger Equation.

For Hydrogen-like atoms (He+ or Li++)

Replace e2 with Ze2 (Z is the atomic number).

Replace m with the reduced mass, m..

Our task is to find the wave functions ª(r; µ; Á) that satisfy this

equation, and hence to find the allowed quantized energies E.

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2. Angular Momentum

Classical definition of angular momentum:

For circular orbits this simplifies to L = mvr, and in Bohr’s model, L

was quantized in integer units of h.

However, the full quantum treatment is more complicated, and requires

the introduction of two other quantum numbers l and ml, as we shall now

see.

The components of L are given by

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Angular Momentum …

In quantum mechanics we know that the linear momentum by

differential operators:

Therefore, the quantum mechanical operators for the angular

momentum are given by:

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Angular Momentum ...

The magnitude of the angular momentum is given by:

Now we therefore have the quantum mechanical operator

It should be noted that this operators should be understood in terms

of repeated operations:

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Angular Momentum … It can be shown that the components of the angular momentum

operator do not commute, that is

However, in fact we can show that:

where the “commutator bracket” [ˆLx; ˆLy] is defined by

Remember that: if two quantum mechanical operators do not

commute, then it is not possible to know their values simultaneously.

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Angular Momentum …

For example, the operators for position and momentum in a one-

dimensional system:

Thus we have:

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3. Solution of the Schrödinger Equation for H

satisfies the azimuthal equation for any value of mℓ.

The solution must be single valued to be a valid solution for any f:

mℓ must be an integer (positive or negative) for this to be true.

Now set the left side equal to −mℓ2 and rearrange it [divide by sin2(q)].

Now, the left side depends only on r, and the right side depends only

on θ. We can use the same trick again!

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Solution of the Schrödinger Equation for H

Set each side equal to the constant ℓ(ℓ + 1).

Radial equation

Angular equation

We’ve separated the Schrödinger equation into three ordinary second-

order differential equations, each containing only one variable.

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Solution of the Radial Equation for H

The radial equation is called the associated Laguerre equation and the

solutions R are called associated Laguerre functions. There are

infinitely many of them, for values of n = 1, 2, 3, …

Assume that the ground state has n = 1 and ℓ = 0. Let’s find this solution.

The radial equation becomes:

The derivative of yields two terms.

This yields:

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Solution of the Radial

Equation for HTry a solution

A is a normalization constant.

a0 is a constant with the dimension of length.

Take derivatives of R and insert them into the radial equation.

To satisfy this equation for any r, both expressions in parentheses

must be zero.

Set the second expression

equal to zero and solve for a0:

Set the first expression equal

to zero and solve for E:

Both are equal to the Bohr results!

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Principal

Quantum

Number n

There are many solutions to the radial wave equation, one for

each positive integer, n.

The result for the quantized energy is:

A negative energy means that the electron and proton are bound

together.

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Quantum Numbers

The three quantum numbers:

n: Principal quantum number

ℓ: Orbital angular momentum quantum number

mℓ: Magnetic (azimuthal) quantum number

The restrictions for the quantum numbers:

n = 1, 2, 3, 4, . . .

ℓ = 0, 1, 2, 3, . . . , n − 1

mℓ = −ℓ, −ℓ + 1, . . . , 0, 1, . . . , ℓ − 1, ℓ

Equivalently:

n > 0

ℓ < n

|mℓ| ≤ ℓ

The energy levels are:

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Hydrogen Atom Radial Wave Functions

First few

radial

wave

functions

Rnℓ

Sub-

scripts

on R

specify

the

values of

n and ℓ.

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Solution of the Angular and Azimuthal

Equations

The solutions to the azimuthal equation are:

Solutions to the angular and azimuthal equations are linked

because both have mℓ.

Physicists usually group these solutions together into

functions called Spherical Harmonics:

spherical harmonics

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Normalized

Spherical

Harmonics

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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 mℓ. The wave function

becomes

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Orbital Angular Momentum Quantum

Number ℓ

Energy levels are degenerate with respect to ℓ (the energy is

independent of ℓ).

Physicists use letter names for the various ℓ values:

ℓ = 0 1 2 3 4 5 . . .

Letter = s p d f g h . . .

Atomic states are usualy referred to by their values of n and ℓ.

A state with n = 2 and ℓ = 1 is called a 2p state.

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Orbital Angular Momentum Quantum

Number ℓ

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

Classically, the orbital angular momentum with L = mvorbitalr.

L is related to ℓ by

In an ℓ = 0 state,

This disagrees with Bohr’s

semi-classical “planetary”

model of electrons orbiting

a nucleus L = nħ.

Classical orbits—which do not

exist in quantum mechanics

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Example: ℓ = 2:

Only certain orientations of are

possible. This is called space

quantization.

And (except when ℓ = 0) we just

don’t know Lx and Ly!

Magnetic Quantum

Number mℓ

The angle f is the angle from the

z axis.

The solution for g(f) specifies that

mℓ is an integer and is related to

the z component of L:

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Rough derivation of ‹L2› = ℓ(ℓ+1)ħ2

We expect the average of the angular momentum components

squared to be the same due to spherical symmetry:

But

Averaging over all mℓ values (assuming each is equally likely):

2 ( 1)(2 1) / 3

n

m

because:

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In 1896, the Dutch physicist Pieter Zeeman

showed that spectral lines emitted by atoms

in a magnetic field split into multiple energy

levels. It is called the Zeeman effect.

Consider the atom to behave like a small magnet.

Think of an electron as an orbiting circular current loop of I = dq / dt

around the nucleus. If the period is T = 2p r / v,

then I = -e/T = -e/(2p r / v) = -e v /(2p r).

The current loop has a magnetic moment m = IA = [-e v /(2p r)] p r2

=

[-e/2m] mrv:

where L = mvr is the magnitude of the orbital angular momentum.

4. Magnetic Effects on Atomic

Spectra—The Zeeman Effect

2

eL

mm

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If the magnetic field is in the z-direction, we only care about the z-

component of m:

where mB = eħ / 2m is called the Bohr magneton.

The potential energy due to the

magnetic field is:

The Zeeman Effect

( )2 2

z z B

e eL m m

m mm m

2

eL

mm

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The Zeeman Effect

A magnetic field splits the mℓ levels. The potential energy is quantized

and now also depends on the magnetic quantum number mℓ.

When a magnetic field is applied, the 2p level of atomic hydrogen is

split into three different energy states with energy difference of ΔE =

mBB Δmℓ.

mℓ Energy

1 E0 + μBB

0 E0

−1 E0 − μBB

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The

Zeeman

Effect

The transition

from 2p to 1s,

split by a

magnetic field.

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An atomic beam of particles in the ℓ = 1 state pass through a

magnetic field along the z direction.

The Zeeman Effect

The mℓ = +1 state will be deflected down, the mℓ = −1 state up, and

the mℓ = 0 state will be undeflected.

( / )Bm dB dzm

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5. Energy Levels

and Electron

Probabilities

For hydrogen, the energy

level depends on the prin-

cipal quantum number n.

In the ground state, an atom

cannot emit radiation. It can

absorb electromagnetic

radiation, or gain energy

through inelastic

bombardment by particles.

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Selection Rules

The probability is proportional to

the mag square of the

dipole moment:

Allowed transitions:

Electrons absorbing or emitting photons can change states

when Δℓ = 1 and Δmℓ = 0, 1.

Forbidden transitions:

Other transitions are possible

but occur with much smaller

probabilities.

*d er

We can use the wave functions

to calculate transition probabilities

for the electron to change from

one state to another.

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Probability Distribution Functions

We use the wave functions to calculate the probability

distributions of the electrons.

The “position” of the electron is spread over space and is not

well defined.

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 .

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Probability Distribution Functions

The differential volume element in spherical polar coordinates is

Therefore,

At the moment, we’re only interested in the radial dependence.

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

and ℓ.

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R(r) and P(r)

for the lowest-

lying states of

the hydrogen

atom.

Probability

Distribution

Functions

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Probability Distribution Functions

The probability density for the hydrogen atom for three different

electron states.

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6. Intrinsic Spin

In 1925, grad students, Samuel

Goudsmit and George Uhlenbeck,

in Holland proposed that the

electron must have an

intrinsic angular momentum

and therefore a magnetic moment.

Paul Ehrenfest showed that, if so, the surface of the spinning

electron should be moving faster than the speed of light!

In order to explain experimental data, Goudsmit and Uhlenbeck

proposed that the electron must have an intrinsic spin

quantum number s = ½.

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Intrinsic Spin

The spinning electron reacts similarly to

the orbiting electron in a magnetic field.

The magnetic spin quantum number ms

has only two values, ms = ½.

The electron’s spin will be either “up” or

“down” and can never be spinning with its

magnetic moment μs exactly along the z

axis.

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Intrinsic Spin

The magnetic moment is .

The coefficient of is −2μB and is a

consequence of relativistic quantum mechanics.

Writing in terms of the gyromagnetic ratio, g: gℓ = 1 and gs = 2:

The z component of .

In an ℓ = 0 state:

Apply ms and the potential energy becomes:

no splitting due to .

there is space quantization due to the intrinsic spin.

and

2L

eL

mm

Recall:

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Multi-electron atoms

When more than one electron is involved, the potential and the

wave function are functions of more than one position:

1 2( , ,..., )NV V r r r

1 2( , ,..., , )Nr r r t

Solving the Schrodinger Equation in this case can be very hard.

But we can approximate the solution as the product of single-

particle wave functions:

1 2 1 1 2 2( , ,..., , ) ( , ) ( , ) ( , )N N Nr r r t r t r t r t

And it turns out that, for electrons (and other spin ½ particles), all

the i’s must be different. This is the Pauli Exclusion Principle.

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Molecules have many energy levels.

A typical molecule’s energy levels:

Ground

electronic state

1st excited

electronic state

2nd excited

electronic state

Energ

y

Transition

Lowest vibrational and

rotational level of this

electronic “manifold”

Excited vibrational and

rotational level

There are many other

complications, such as

spin-orbit coupling,

nuclear spin, etc.,

which split levels.

E = Eelectonic + Evibrational + Erotational

As a result, molecules generally have very complex spectra.

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Generalized Uncertainty Principle

Define the Commutator of two operators, A and B:

,A B AB BA

Then the uncertainty relation between the two corresponding

observables will be:

*12

,A B A B So if A and B commute, the two observables can be measured

simultaneously. If not, they can’t.

Example: ,p x px xp i x x ix x

xi i x x i i

x x x

,p x i So: / 2p x and

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Two Types of Uncertainty in Quantum

Mechanics

We’ve seen that some quantities (e.g., energy levels) can be

computed precisely, and some not (Lx).

Whatever the case, the accuracy of their measured values is

limited by the Uncertainty Principle. For example, energies can

only be measured to an accuracy of ħ /t, where t is how long

we spent doing the measurement.

And there is another type of uncertainty: we often simply don’t

know which state an atom is in.

For example, suppose we have a batch of, say, 100 atoms,

which we excite with just one photon. Only one atom is excited,

but which one? We might say that each atom has a 1% chance

of being in an excited state and a 99% chance of being in the

ground state. This is called a superposition state.

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Superpositions of states

Stationary states are stationary. But an atom can be in a

superposition of two stationary states, and this state moves.

1 1 1 2 2 2( , ) ( )exp( / ) ( )exp( / )r t a r iE t a r iE t

2 2 2

1 1 2 2

*

1 1 2 2 2 1

( , ) ( ) ( )

2Re ( ) ( )exp[ ( ) / ]

r t a r a r

a r a r i E E t

where |ai|2 is the probability that the atom is in state i.

Interestingly, this lack of knowledge means that the

atom is vibrating:

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Superpositions of states

E

nerg

y

Ground level, E1

Excited level, E2

E = hn

The atom is at least partially in

an excited state.

The atom is vibrating

at frequency, n.

Vibrations occur at the frequency difference between the two levels.

2 2 2

1 1 2 2

*

1 1 2 2 2 1

( , ) ( ) ( )

2Re ( ) ( )exp[ ( ) / ]

r t a r a r

a r a r i E E t

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Calculations in Physics: Semi-classical

physics

The most precise computations are performed fully quantum-

mechanically by calculating the potential precisely and solving

Schrodinger’s Equation. But they can be very difficult.

The least precise calculations are performed classically,

neglecting quantization and using Newton’s Laws.

An intermediate case is semi-classical computations, in

which an atom’s energy levels are computed quantum-

mechanically, but additional effects, such as light waves, are

treated classically. Our Thomson Scattering calculation of a and

n was an example of this.