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.

White light (all wavelengths) passed

through hydrogen gas, then analyzed with a

spectrometer.The dark lines in the absorption spectrum are

at the exact same wavelengths as the

colored bright lines in the emission spectrum.

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 hydrogen given by ;– 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 ⇒amount;

• 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 by

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

box through

The ‘electron in a box’ model

Therefore

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’ modelThe electron’s Ek can only be

• 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

given by ; where n is an integer that represents the energy level the electron inhabits and C is a constant equal to ; 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.

• ; theory predicts that the electron in the hydrogen atom has quantized energy

The Schrӧdinger Theory

• 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)

ExampleShow how the formula for the electron energy in the Schrӧdinger theory can be used to derive the

empirical Balmer formula mentioned earlier .

ExampleCalculate the wavelength of the photon emitted in

the transition from n=3 to n=2.

The Schrӧdinger TheoryThe 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.

• ; 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 a = opening size = uncertainty in position = Δx. From the figure

The Heisenberg Uncertainty PrincipleApplication: 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 must be and . Applying this to an electron in the

hydrogen atom (L≈10-10m): Which is the correct order of magnitude value of

the electron’s kinetic energy.

The Heisenberg Uncertainty Principle

Note the resemblance of this formula () to the formula for the energy obtained earlier in the ‘electron in a box’ model (). 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

• ; ΔE – uncertainty in energy, Δt – uncertainty in time, h = planck’s constant

Assignment

• 8,11,15