Quantum Mechanics

#### Outline an experiment to test the Einstein model

When light strikes a metal some of the electrons in the metal can be “knocked” off an atom and can fly away or dissociate from the atom. If enough light strikes the metal a substantial current can flow. This is the basis for what is called the photoelectric effect.

When the experiment is done a few surprising results are found:

1. The electrons are released immediately.
2. Increasing the intensity (the amount) of light hitting the metal increased the current or number of electrons released but did not affect the (maximum) kinetic energy of the electrons.
3. If the frequency of the incident light is lowered, at a certain frequency the current stops flowing no matter how intense the light.

These results are incompatible with the wave model of light. This discrepancies were well know and people were working on it, but Einstein was the first to fully explain the what was going on, and he won the Nobel Prize for it.

#### First a little background:

There is a phenomena called blackbody radiation. Any body will radiate light, the frequency of the light is distributed over a range of frequencies. The distribution shifts as the temperature of the body changes. The “classical” theory provided an acceptable explanation at lower frequencies, however at higher frequencies classical theory predicted that the body would radiate an infinite amount of energy… clearly impossible. A new theory was needed.

A guy by the name of Max Planck provided the new theory. The key to Planck’s theory was that the energy of light was quantized, that is it could only take on certain energy levels. He hypothesized that the energy of the light was given by the equation:

(1)
$$E=h f$$

Were h is now known as Planck’s constant and f is the frequency of the light, this equation is in the IB formula booklet. Planck’s theory provided a theoretical framework that explained the experimental observations and quantum mechanics was started…

#### Back to the photoelectric effect:

Einstein used Planck’s theory that the energy of light was quantized to explain the photoelectric effect. An experiment similar to that shown to the right can be done to test the properties of the photoelectric effect.

If the frequency of the light is low and slowly increased it will be found that at some frequency a current will start to flow, this is called the threshold frequency. If the frequency of the light is well above the threshold frequency, a negative potential can be applied in the cathode tube, as the potential is increased the current decreases until the current stops, the voltage that the current stops flowing at is called the cut-off voltage (or the stopping voltage). Therefore we can say the maximum energy of any of the electrons is equal to:

(2)
$$E = eV_s$$

This provides a method to measure the maximum energy of the electrons that are given off by the metal for a given frequency of light. If the frequency of light is increased the current will start to flow again, thus by increases the frequency of light we have increased the maximum kinetic energy of the electrons. If we plot the maximum kinetic energy versus light frequency we find the following.

The x-intercept is the cut-off frequency, or minimum frequency of the light required to release electrons from the metal. The slope of the line is equal to h, Planck’s constant. The y-intercept, represents the amount of energy needed to remove the electron from the atoms, or the ionization energy.

$\phi$ is called the work function, how much work has to be done to release the ionize the atom. The work function is different for different metals. From the graph we can write a function for the energy of the incident light:

(3)
\begin{align} E = E_{K \: max} + \phi \end{align}

We also can describe the energy of the photon in terms of Planck’s constant and the frequency.

(4)
\begin{align} hf = E_{K \: max} + \phi \end{align}

This equation is in the IB formula booklet. If the incident light frequency is the cut-off frequency then the electron will have no kinetic energy:

(5)
\begin{align} E_{light} = hf_0 = \phi \end{align}

This is the minimum energy the light needs to have in order to generate a current. The idea of a minimum energy does not match up with a wave model of light…

We can relate the energy of the incident light to the cut-off frequency and the maximum kinetic energy of the electrons:

(6)
$$hf = hf_0 + eV_s$$

Yet another equation in the IB formula booklet!

With a wave model of light, light is continuous and should be able to continuously give energy to the metal, and thus an electron would be released when enough energy was given to the metal. Einstein proposed the theory that light is not a wave, but is a particle with quantized energy, or at the very least that light had both wave and particle like properties. The particles are called photons, if the incident photon has enough energy to knock an electron off, then it is absorbed and an electron is released. If the energy of the photon is too low then the photon is reflected or transmitted…

#### Describe de Broglie’s hypothesis and the concept of matter waves.

After Einstein showed that light had particle like properties, a guy by the name of de Broglie began to wonder if matter, things normally thought of as particles, had wave like properties…

The energy of a particle is given by the equation:

(7)
\begin{align} E = \sqrt{p^2 c^2 + m_0^2 c^4} \end{align}

If the object has zero rest mass then the energy of the object is:

(8)
$$E = pc$$

Therefore for light:

(9)
$$E = hf = pc$$

Solving for momentum:

(10)
\begin{align} p = \frac{hf}{c} \end{align}

The speed of light is defined as:

(11)
\begin{align} c = \lambda F \end{align}

Therefore the momentum can be described as:

(12)
\begin{align} p = \frac{h}{f} \end{align}

De Broglie hypothesized that other particles with momentum may have a wavelength as well. The so-called de Broglie wavelength is:

(13)
\begin{align} \lambda = \frac{h}{p} \end{align}

De Broglie did not have substantial experimental data to justify this conclusion it was a hypothesis.

#### Outline an experiment to test the de Broglie hypothesis.

A few years after de Broglie made his hypothesis three physicist (Davisson, Germer, Thomson) independently performed experiments that supported de Broglie’s hypothesis.

A beam of low energy electrons was aimed at different angles at a nickel crystal. The electrons appeared to reflect (bounce) off the nickel. However they found that at certain angles the electrons did not appear to bounce off. What they realized was the pattern was the same as for light passing through a diffraction grating. It appeared that the electrons were interfering with themselves of each other…

If low energy (low momentum) electrons are shot one at a time at a double slit an interference pattern can be detected. But for an interference pattern to form the electrons must behave as a wave. Also if the electrons are shot one at a time, then they must pass through both slits and interfere with themselves!

To make the situation stranger yet…

It seems impossible for an electron to pass through two slits at the same time, surely it must go through one or the other. If a detector is set up at one of the slits, to see if it goes through that slit, the interference pattern is destroyed… Simply by testing or measuring which slit it goes through forces the electron to go through only one slit! This can be at least somewhat explained by the Heisenberg uncertainty principle.

#### Explain how atomic spectra provide evidence for the quantization of energy in atoms

Umm, well, gee, I guess you could use a spectroscope…

Essentially light needs to reflect off of a diffraction grating, this splits the light into individual wavelength or frequencies (this is because light of different wavelength refracts at slightly different angles). These can then be viewed and analyzed.

Atomic spectra are not continuous, they are discrete. That is the light given off is only particular wavelengths and is characteristic of the atom giving off the light. The frequency of the light is discrete or quantized and the energy of the light is dependent on the frequency, this provides further evidence for the quantization of energy.

Examples of atomic spectra (visible light only):

Each element has its own characteristic spectra. Notice that the number of lines increases with the size or complexity of the atom. Each line (color) is caused be the transition of an electron from an excited state to a less excited state. The more electrons the atom has the more possible combinations and therefore more lines in the spectra.

#### Outline the Bohr model of the hydrogen atom

Niels Bohr in 1913 developed a model of the hydrogen atom that was able to explain the emission and absorption spectra of hydrogen. His model assumed that there were special orbits that an electron could be in and would not radiate. In his model the electrons literally orbited the nucleus in the same way that a planet orbits a star. The orbits were quantized in terms of their allowable angular momentum (rotational momentum). Therefore the radii and the energy of the orbits is also quantized. The allowable energy for the orbits is:

(14)
\begin{align} E \propto \frac{1}{n^2} \end{align}

Where n is the orbit number, being the lowest and most stable orbit. The energy of the orbit is the energy required to ionize (or remove) the electron, an electron in orbit is defined to have negative potential energy (exactly like negative gravitational potential energy). When the electrons are excited they jump to a higher energy orbit, eventually (actually very quickly) the electrons drop back down to a more stable orbit and release energy in the form of light (E&M radiation). The energy of the light released is equal to the difference in energy of the two orbits.

For example:

If an electron drops from an n = 2 orbit to an n = 1 orbit the energy released is the difference in the energy:

(15)
$$E = E_2 - E_1$$

In the case of hydrogen the energy in the orbits is equal to:

(16)
\begin{align} E = \frac{k}{n^2} \: ,k=13.6eV \end{align}

Therefore the energy released in the transition is:

(17)
\begin{align} E = \frac{13.6ev}{2^2}-\frac{13.6eV}{1^2}=10.2eV \end{align}

Converting to Joules:

(18)
\begin{align} 10.2eV \left ( \frac{1.6 \times 10^{-19}J}{1eV} \right ) = 1.632 \times 10^{-18}J \end{align}

The frequency of light is given by:

(19)
\begin{align} f=\frac{E}{h} = \frac{1.632 \times 10^{-18}J}{6.64 \times 10^{-34}J \cdot s^-1} = 2.46 \times 10^{15} Hz \end{align}

Which is gives a wavelength of approximately:

(20)
\begin{align} \lambda = \frac{c}{f}=1.22nm \end{align}

Which is an infrared wavelength.

The reverse also happens when light is incident on an atom and has the correct frequency (energy) the light is absorbed and the electron is excited into a higher orbital.

##### Not an experiment, but some awesomeness about quantum mechanics.

The reason this happens is, as said, that light is quantized! Note how even with the weird blinking the data fits with well known "classical physics" laws. Got to love this stuff.

#### State the limitations of the Bohr model

In the Bohr model the electron is assumed to be orbiting the nucleus like a planet which means that it is continuously accelerating. Any accelerating electric charge will emit E&M radiation. In the Bohr model the orbits are special and it is assumed that the electrons do not radiate while in the special orbits. There is no physical reason or justification, it simply makes the model work. The model only works for hydrogen, it can not explain more complicated atoms. There is no justification or explanation of the quantization of angular momentum… By constricting the electron to a known orbit, Bohr’s model is also in violation of Heisenberg’s uncertainty principle.

It explains a lot, but it is not a complete model, a new model is needed.

#### Outline the Schrödinger model of the hydrogen atom

A new proposed model for the hydrogen atom was created by Erwin Schrödinger. His model used de Broglie’s hypothesis that particles have wave properties. He proposed that the electron formed a standing wave around the nucleus. Each orbit corresponded to an integer number of wavelengths, i.e. the n=1 orbit is one wavelength in circumference, the n=2 orbit is two wavelengths in circumference, etc.

Schrödinger’s model used his wave equation, which is used to describes the properties of the particle. The square of the amplitude of the wave equation represents the probability of finding the particle in a given location. Schrödinger’s model no longer had the electrons in a given location but can only describe the probability of the electron being found in a given location! The fact that the location of the electron is not known is in agreement with the uncertainty principle.

For the hydrogen atom Schrödinger’s model predicts a high likelihood that the electron will be in a spherical orbit, which matches up with Bohr’s model. However Schrödinger is able to explain more complex atoms and has better justification for the model.

#### Explain the origins of the features of a typical X-ray spectrum

X-rays can be produced by shooting electrons at a metallic target. As the electrons collide with the target the (de) accelerate, thus they emit light. The frequency of the light is dependent on the kinetic energy of the electrons. A high electric potential is used to accelerate the electrons, thus the frequency of the light is dependent on the potential difference.

In the diagram shown to the right a filament is heated until the electrons are emitted, the electrons are then accelerated through a electric potential until they strike a tungsten target.

A typical X-ray spectrum is shown to the below. There are two notable features. First there is a continuous spectrum, this is produced by the acceleration of the electrons. The range of wavelengths/frequencies is only a function of the electric potential.

The spikes are due to emissions from the target. Occansionally the electrons strike the target with the correct amount of energy to excite an orbiting electron, when the electrons falls back down X-rays are emitted in the characteristic peaks (frequencies).