Transcript 7.5 Quantum Theory & the Uncertainty Principle

7.5 Quantum Theory & the
Uncertainty Principle
“But Science, so who knows?!”
Objectives
• Describe emission and absorption spectra and
understand their significance for atomic
structure;
• Explain the origin of atomic energy levels in terms
of the ‘electron in a box’ model;
• Describe the hydrogen atom according to
Schrӧdinger;
• Do calculations involving wavelengths of spectral
lines and energy level differences;
• Outline the Heisenberg uncertainty principle in
terms of position-momentum and time-energy.
Atomic spectra
• Hydrogen gas heated to high temps/exposed
to high electric field ⇒ glows (EMITS light)
• Analyze light by
sending it through
a spectrometer
(splits light into
component
wavelengths)
• λ = 656 nm (Hα) red
• λ = 486 nm (Hβ)
blue-green
Emission spectrum – the spectrum of light
that has been emitted by a gas
Bright Lines
Absorption Spectrum – the spectrum of light
that has been transmitted through a gas.
The dark lines in the
absorption spectrum are
at the exact same
wavelengths as the
colored bright lines in the
emission spectrum.
White light (all
wavelengths) passed
through hydrogen gas,
then analyzed with a
spectrometer.
Atomic Spectra
• Emission & absorption lines at specific
wavelengths for a particular gas
• Scientific community: ?????
• 1885 – Johann Balmer (accidentally) discovers
wavelengths in the emission spectrum of
1
λ
hydrogen given by = 𝑅
1
1
− 2
4
𝑛
– n is an integer (3,4,5…)
– R is a constant
– Scientific community STILL ?????
;
Atomic Spectra
• Light carries energy ⇒ reasonable (based on
conservation of energy) to assume that the emitted
energy is equal to the difference between the total
energy of the atom before and after emission
• Emitted light consists of photons of a specific
wavelength ⇒ emitted energy must be a specific
ℎ𝑐
amount; 𝐸 = ℎ𝑓 =
λ
• Therefore, the energy of an atom is discrete (not
continuous) but how could this be? How must our
model change in light of this new evidence?
The ‘electron in a box’ model
Imagine that an
electron is contained
within a “box” of linear
size L. The electron,
treated as a wave,
according to de Broglie,
has a wavelength
associated with it given
ℎ
by λ =
𝑝
Since the electron is
confined to the box, it is
reasonable to assume that
the electron wave is zero at
both edges of the box.
The ‘electron in a box’ model
In addition, since the
electron cannot lose
energy, it is also
reasonable to assume
that the wave
associated with the
electron in this case is
a standing wave.
So we want a standing wave
that will have nodes at x = 0
and x = L. This implies that
the wavelength must be
related to the size L of the
2𝐿
box through λ =
𝑛
The ‘electron in a box’ model
ℎ
ℎ
ℎ
𝑛ℎ
λ= →p= =
=
2𝐿
𝑝
λ
2𝐿
𝑛
Therefore
2
𝑛ℎ
𝑝2
𝑛2 ℎ2
2𝐿
𝐸𝐾 =
=
=
2𝑚
2𝑚
8𝑚𝐿2
This result shows that, because we treated the
electron as a standing wave in a ‘box’, we deduce that
the electron’s energy is ‘quantized’ or discrete, i.e. it
cannot have any arbitrary value.
The ‘electron in a box’ model
The electron’s Ek can only be
ℎ2
𝐸𝐾 = 1 ×
𝑛=1
2
8𝑚𝐿
ℎ2
𝐸𝐾 = 4 ×
𝑛=2
2
8𝑚𝐿
ℎ2
𝐸𝐾 = 9 ×
𝑛=3
2
8𝑚𝐿
• This model gives us a discrete set of energies
• Not a realistic model for an electron in an atom, but it
does show the discrete nature of the electron energy
when the electron is treated as a wave; points toward
the correct answer.
The Schrӧdinger Theory
• 1926 – Austrian Physicist Erwin Schrӧdinger
• Assumes as a basic principle that there is a wave
associated to the electron (like de Broglie), called
the wavefunction, ψ(x,t).
• The wavefunction is a function of position x and
time t.
• Given the force that act on an electron, it is
possible, in principle, to solve a complicated
differential equation obeyed by the wavefunction
(the Schrӧdinger equation) and obtain ψ(x,t).
The Schrӧdinger Theory
• For example, there is one wavefunction for a free
electron, another for an electron in the hydrogen
atom, etc.
The interpretation of what ψ(x,t) really
means came from German physicist Max
Born. He suggested that |ψ(𝒙,𝒕)|𝟐 (the
square of the absolute value of ψ(x,t) can
be used to find the probability that an
electron will be found near position x at
time t.
The Schrӧdinger Theory
• The theory only gives probabilities for finding an
electron somewhere – it does not pinpoint and
electron at a particular point in space; a radical
change from ordinary (classical) physics where
objects have well defined positions.
• When the Schrӧdinger theory is applied to the
electron in a hydrogen atom it gives results similar
to the simple electron in a box example of the
previous section.
The Schrӧdinger Theory
• It predicts that the total energy of the electron is
𝐶
given by 𝐸 = − 2; where n is an integer that
𝑛
represents the energy level the electron inhabits
𝟐𝝅𝟐 𝒎𝒆𝟒 𝒌𝟐
𝒉𝟐
and C is a constant equal to 𝑪 =
=
𝟏𝟑. 𝟔 𝒆𝑽; k is the constant in Coulomb’s law, m is
the mass of the electron, e is the charge of the
electron and h is Planck’s constant.
−𝟏𝟑.𝟔 𝒆𝑽
;
𝟐
𝒏
• 𝑬=
theory predicts that the electron in
the hydrogen atom has quantized energy
The Schrӧdinger Theory
−𝟏𝟑.𝟔 𝒆𝑽
𝒏𝟐
• 𝑬=
• High n energy levels are
very close together
• When an electron absorbs
a photon, it jumps up an
energy level (absorption
spectra)
• When the electron loses
enough energy to drop
down one or more levels, it
emits a photon of energy
equal to the energy it lost
(emission spectra)
Example
Show how the formula for the electron energy in
the Schrӧdinger theory can be used to derive the
1
empirical Balmer formula mentioned earlier =
𝑅
1
4
−
1
𝑛2
λ
.
Example
Calculate the wavelength of the photon emitted in
the transition from n=3 to n=2.
The Schrӧdinger Theory
The variation of the probability distribultion function
(pdf) with distance r from the nucleus for the n=1
(lowest) energy level of the hydrogen atom. The
height of the graph is proportional to |ψ(𝒙,𝒕)|𝟐.
The shaded area
is the probability
for finding the
electron at a
distance from the
nucleus between
r=a and r=b.
Assignment
• 3,4,5,7
The Heisenberg Uncertainty Principle
• Discovered 1927 - Named after Werner
Heisenberg (1901 – 1976); one of the
founders of quantum mechanics
• Founding idea: wave-particle duality –
particles sometimes behave like waves
and waves sometimes behave like
particles, so that we cannot cleanly divide
physical objects as either particles or
waves
The Heisenberg Uncertainty Principle
• The Heisenberg uncertainty principle applied to
position and momentum states that it is not
possible to measure simultaneously the position
and momentum of something with indefinite
precision – representative of a fundamental
property of nature; nothing to do with equipment.
𝒉
;
𝟒𝝅
• ∆𝒙∆𝒑 ≥
Δx – uncertainty in position, Δp –
uncertainty in momentum, h = planck’s constant
• Making momentum as accurate as possible makes
position inaccurate. If one is zero, the other is
infinite.
The Heisenberg Uncertainty Principle
• Imagine: electrons emitted from a hot wire in a
cathode ray tube (crt) and we try to make them
move in a horizontal straight line by inserting a
metal with a small opening of size a. we can
make the electron beam as thin as possible by
making the opening as small as possible –
electrons must be somewhere within the
opening so Δx < a.
• a should not be on the same order of magnitude
as the de Broglie wavelength of the electrons to
avoid diffraction.
The Heisenberg Uncertainty Principle
• Here, too, the electron will diffract through the
opening ⇒ some electrons emerge in a
direction that is no longer horizontal.
• We can describe this phenomenon by saying
that there is an uncertainty in the electron’s
momentum in the vertical direction of
magnitude Δp
The Heisenberg Uncertainty Principle
• The angle by which the electron is diffracted is
given by 𝜃 ≈
λ
;a
𝑎
= opening size = uncertainty in
position = Δx. From the figure 𝜃 ≈
•
λ
𝑎
≈
∆𝑝
𝑝
⇒ ∆𝑝∆𝑥 ≈ λ𝑝 = ℎ
∆𝑝
𝑝
The Heisenberg Uncertainty Principle
Application: consider an electron which is known to
be confined within a region of size L. Then the
uncertainty in position must satisfy Δx < L, so Δp
ℎ
ℎ
𝑝2
∆𝑝2
must be ∆𝑝 ≈
≈
and 𝐸𝑘 =
>
≈
ℎ2
16𝜋2 𝑚𝐿2
4𝜋∆𝑥
4𝜋𝐿
2𝑚
. Applying this to an electron in the
hydrogen atom (L≈10-10m): 𝐸𝑘 ≈
6.63×10−34 𝐽
16𝜋2
2𝑚
9.1×10−31 𝑘𝑔
2
10−10 𝑚 2
ℎ2
16𝜋2 𝑚𝐿2
−19
= 3 × 10
=
𝐽 ≈ 2𝑒𝑉
Which is the correct order of magnitude value of
the electron’s kinetic energy.
The Heisenberg Uncertainty Principle
Note the resemblance of this formula (𝐸𝑘 ≈
ℎ2
)
2
2
16𝜋 𝑚𝐿
to the formula for the energy obtained
𝑛2 ℎ 2
).
2
8𝑚𝐿
earlier in the ‘electron in a box’ model (𝐸𝐾 =
Apart from a few numerical factors (of order 1) the
two are the same, indicating the basic connection
between the uncertainty principle and duality.
The Heisenberg Uncertainty Principle
• Also applicable to energy and time.
• The Heisenberg uncertainty principle
applied to energy and time states that it
is not possible to know simultaneously
the energy and time of something with
indefinite precision
𝒉
;
𝟒𝝅
• ∆𝑬∆𝒕 ≥
ΔE – uncertainty in energy,
Δt – uncertainty in time, h = planck’s
constant
Assignment
• 8,11,15