Download Applying the Concepts of Matter Waves The Heisenberg Uncertainty

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Applying the Concepts of Matter Waves
Once the concept of matter waves was advanced, it was quite
easy to rationalize the ad hoc quantization of angular momentum
that Bohr had introduced: stationary states occurred when
an integral number of deBroglie waves could fit exactly on the
circumference of the orbit:
2πR = n
h
mv
n = 1, 2, 3, 4, …
Panels (a) and (b) show cases where 4 or 5 deBroglie waves fit exactly. We say that standing
waves corresponding to complete constructive interference are formed. However, when we
attempt to fit a non-integral multiple of deBroglie wavelengths on the circle, as in panel (c),
complete destructive interference occurs quickly.
The conclusion: stationary states are a consequence of constructive interference of
matter waves in a fixed region of space. This was the conceptual foundation for the
“New Quantum Theory”, Schrödinger’s wave mechanics.
The Heisenberg Uncertainty Principle
Ascribing the properties of waves to matter comes at a price. It is a fundamental property of
waves that it is impossible to determine the position of a wave and its momentum simultaneously
with arbitrarily high precision. This is true for light waves as well as matter waves. When this
idea is applied to matter waves, the Heisenberg Uncertainty Principle results:
Let ∆x be the uncertainty in the measurement of the position of a particle. Let ∆p be the
corresponding uncertainty in the measurement of momentum of a particle:
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Then, the Heisenberg Uncertainty Principle states that
h
∆p∆x ≥
2
We can apply this idea to the circular orbits in the Bohr atom. Effectively, the Uncertainly
Principle means that we cannot speak of the electron’s motion in terms of a well-defined circular
trajectory with a precise radius. The electron’s motion is “fuzzy”, and all we can do is talk about
the probability of finding the electron in a region of space.
An interesting example:
So, what are some other implications? As an example, consider the following: we’ll learn later
in this lecture that there is a finite probability of finding the electron inside the nucleus of a
hydrogen atom. Can we actually prove by direct detection that the electron is in the nucleus?
If the electron is in the nucleus, then ∆x is about 10-15 m. So, the uncertainty in the electron’s
momentum is
h
= (1.054 × 10-34 J sec)/[2 × 10-15 m] = 5.3 × 10-20 kg m/sec
∆p ≥
2∆x
If this is the uncertainty in the electron’s momentum, then the uncertainty in the electron’s
kinetic energy is (using E = p2/2m)
(∆p)2/2m = (5.3 × 10-20)2/[2 × 9.1 × 10-31 ] J = 1.5 × 10-9 J
Remember that the energy required to ionize hydrogen ( to remove the electron
completely from the atom) is 2.178 × 10-18 J.
So, the uncertainty in the electron’s kinetic energy resulting from confining the electron
in the nucleus is one billion times larger than the ionization energy. So, it would be
impossible to make a measurement that observes the electron in the nucleus. If you
attempted the measurement, the electron would leave the atom entirely.
The Wave Equation for Matter Waves
A wave equation is a differential equation that describes the amplitude of the wave over all
space. That is, the wave equation involves derivatives of the wave amplitude. Wave equations
for light that correctly described its wave nature were known even in the 19th century. Erwin
Schrödinger was the first to take the classical wave equation and incorporate the deBroglie
wavelength concept into it. The resultant equation, the Schrödinger equation, (logically
enough!) describes the amplitude of a matter wave through all space.
The equation has the following form:
(Kinetic energy operator + potential energy operator)ψ = Eψ
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The function ψ is the “wavefunction”, the amplitude of the matter wave. The operator for
kinetic energy has second derivatives in it, so the wave equation is a second-order differential
equation. Because the equation allows for the incorporation of the “true” potential energy of the
system, atoms with more than one electron can be accommodated, unlike the Bohr theory.
The wavefunction is interpreted as follows:
⎢ψ(x) ⎢2 dx = probability of finding the particle in the interval from x to x + dx
The way we will use this idea in atoms and in molecules is to think about the electron density
throughout space.
The solutions to the Schrödinger equation exist only for certain allowed values of the energy. It
is only for these “allowed” values that matter waves undergo constructive interference. For all
other energies, the matter waves experience complete destructive interference. The process is
analogous to fitting an integral number of deBroglie waves in a fixed region of space, like the
figures at the beginning of this lecture.
The Hydrogen Atom
We are now in a position to describe “what the electron is doing” in the hydrogen atom. Choose
a coordinate system with the proton at the origin. The potential energy depends only on the
distance of the electron from the origin: V(r) = -e2/4πε0r. So, let’s use spherical polar
coordinates, as shown in the diagram. We can specify the position of the electron wither in
Cartesian coordinates (x, y, z) or (r, θ, φ)
We can “trig out” the transformation:
z = rcosθ
x = rsinθcosφ
y = rsinθsinφ
Note that (x2 + y2 + z2)1/2 = r
The condition that standing matter waves be formed in three
spatial dimensions (x, y, z) leads to three quantum numbers.
First of all, we find that the energy of the system depends only on a single quantum number, n,
the principal quantum number. Just like the Bohr atom,
⎛ mZ 2 e 4
E n = −⎜⎜ 2 2
⎝ 8ε 0 h
⎞ 1
⎟
⎟ 2
⎠n
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The principal quantum number has the range n = 1, 2, 3, ….. Associated with the angular
coordinates θ and φ are two quantum numbers denoted l , the orbital angular momentum
quantum number, and m l , the magnetic quantum number.
The allowed range of quantum numbers:
n:
1, 2, 3, …..
l:
0, 1, 2, …. n - 1
ml :
-l
, - l +1, …, 0, …+ l
There are 2 l + 1 possible values of the magnetic
quantum number.
Remember that the energy of the atom depends
only on n, so the diagram at the left holds. We say
that levels with the same value of n but different
values of l and m l are degenerate.
We assign a letter to correspond with each value of l :
Value of l
0
1
2
3
Symbol
s
p
d
f
Origin
“sharp”
“principal”
“diffuse”
“fine”
The column labeled “Origin” refers to the original spectroscopic nomenclature assigned to
spectral lines with this characteristic.
Properties of Radial Functions
Let’s look at function with l = 0, the s-orbitals. Table 12.1 gives us the form. For n = 1, we can
only have l = 0, and m l = 0.
1 ⎛ Z
⎜
ψ 1s =
π ⎜⎝ a 0
⎞
⎟⎟
⎠
3/ 2
e − Zr / a 0
Remember that Z = 1 for the hydrogen atom.
a0 = the Bohr radius = 0.529 × 10
-10
m=
ε0h 2
πme 2
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A plot of this function appears in panel b), and a density representation in panel a)
You can see that the density electron density maximizes at the nucleus. Think back to the
example with the Uncertainty Principle. Another useful way of thinking about the electron
density is to calculate how much of the density shown in panel a) can be assigned to concentric
spherical shells:
Maximum at r = a0
This calculation is a product of the square of the wavefunction ⎢ψ1s ⎢2 times the volume of the
spherical shell of radius r and thickness dr. This volume is 4πr2dr. The resultant radial
probability density is the best way to represent the electron density, panel b). The maximum in
this function occurs at 0.529 × 10-10 m, exactly the distance of the first Bohr radius!!
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As we go to the 2s and 3s functions, we see something new occur, as indicated in the next figure:
Note the radial nodes in the n = 2
and 3 functions
There are a number of points that can be made from looking at this figure.
First, the number of nodes increases with n. There are n – 1 radial nodes for an
“ns” function. The number of nodes correlates directly with the energy of the orbital.
Second, there is a shell structure that arises from the local maxima that occur
for larger values of n. This is analogous to the shell structure of the Bohr model.
But note that the electron density is “blurred” out. We can only speak of the
probability of finding the electron in space.
The electron density in s-orbitals is spherically symmetric - like a basketball