The Theoretical origins of the Uncertainty Principle

As can be seen in the previous section (rules 1 + 2 below), many of the 'rules' of Quantum Mechanics are counter intuitive. Another of these counter intuitive rules can be seen as the Uncertainty Principle.

The three rules of Quantum Mechanics can be summed up as follows :-

1. Transitions between states are discrete. There are no such things as intermediate states and as such nothing to describe in this regard.

2. Depending on conditions, matter can behave as either a wave or a particle, on a Quantum level, there is no distinction between them.

3. Every physical system can be modeled mathematically as a wave-function. A wave-function does not physically describe an object, it is a description of all possibilities of that object within that situation.

Note - As with all aspects of Quantum theory it must be remembered that, though counter intuitive, all the discussed Quantum effects can be seen experimentally in the real world.

Wave-Functions

In Quantum Theory, it is not possible to precisely state all conditions at an instant, only their probabilities. For example, it is not possible to state when a particular nucleus will decay radioactively, but it is possible to calculate the probability of a particular nucleus decaying within a certain time.

By extending our understanding of wave-particle duality of the electron to any matter particle it is possible to describe all matter in the form of a wave. In order to do this, however, a clearer description of that wave must be used.

The wave nature of a particle used within Quantum Theory is a ' Wave Packet'. A wave packet is formed by combining together a range of frequency waves to create a short pulse, as is seen in the following diagram :-

Where's the wave ?

Looking at a single wave, its Frequency can be known extremely accurately and its position can be thought of a being anywhere along its length. This means that its position is very imprecise.

Looking at a wave packet instead, the position of the wave is much more defined, but its frequency is much more imprecise.

This gives an example of the Uncertainty Principle, it is possible to measure Frequency or position to a high level of accuracy, but not both at the same time. In fact, even the act of making these observations can interfere with the results.

The Uncertainty Principle

Heisenberg's Uncertainty Principle in relation to the position and Momentum of an object states that:-

Where :-

Δx = The Uncertainty in the position

Δp_{x} = The Uncertainty in the Momentum in the x-direction

h = Plank's Constant ( 6.63x10^{-34} m^{2 }kg s^{-1} )

In order to understand how the above formula can be applied to a particle interaction, the following thought experiment can be used :-

Heisenberg also derived an Uncertainty relationship for Energy and time :-

Where :-

ΔE = The Uncertainty in the Energy of a particle

Δt = The Uncertainty in the time measured

h = Plank's Constant ( 6.63x10^{-34} m^{2 }kg s^{-1} )

Example 1 -

Some of the limitations of the Bohr Model of the Atom are explained by Quantum Mechanics. The position of an electron in orbital n = 1 in the diagram above was measured with an uncertainty of 0.15nm.

Calculate the minimum uncertainty in its momentum.

Example 2 - 2017 Past Paper Q 7a

Laser light is often described as having a single frequency. However, in practice a laser will emit photons with a range of frequencies. Quantum Physics links the frequency of a photon to its energy. Therefore the photons emitted by a laser have a range of energies (ΔE). The range of photon energies is related to the lifetime (Δt) of the atom in the excited state.

By considering the Heisenberg uncertainty principle, state how the lifetime of atoms in the excited state in the neodymium:YAG laser compares with the lifetime of atoms in the excited state in the argon ion laser, justifying your answer.

Model Answer :

Atoms in the Nd:YAG have a shorter lifetime (in the excited state) OR Atoms in the Ar have a longer lifetime (in the excited state). This is because :-

The Uncertainty Principle : Quantum Tunneling

The Uncertainty Principle allows for some unusual effects, and allows particles to "break" some classical mechanics principles. Again, however, it must be noted that most of these can be verified experimentally.

To start with, a classical mechanics view of a potential well ( as seen in Unit 1 ) is required.

In the above diagram, a ball rests at the bottom of a 'dip'. In order for the ball to move out of the dip, it must be given Energy equal to the potential energy at the top of the dip ( E_{p} = mgh ). This is analogous to an all potential wells seen within the Advanced Higher course. A particle must at minimum have Energy equal to the Potential Energy of the Well to escape.

In Quantum Mechanics, this is a little more complicated. Again the above analogy can be used, but in this case it is an Electron within an Electric Potential Well. The diagram below shows the classical understanding :-

In the above diagram, the Electron would require an Energy equal to Ep in order to move over the potential barrier to reach the new location (dotted circle). If the Energy is less than this, the Electron is confined within the Potential Well.

Note - In the above diagram, the Electron is modeled as a particle.

If Quantum Theory is taken into account, however, the electron can be modeled as a Wave-function, and it can be shown that this wave-function upon reaching the barrier, doesn't instantly end, but tapers off quickly. In effect, it passes into the barrier, losing amplitude the further into the barrier it passes. Given a thin enough barrier ( less than ~3 nm ), the amplitude will be non-zero at the far edge of the barrier. This means that there is a finite probability that the Electron could be found outside of the Well.

The diagram below shows this effect visually :-

As can be seen in the diagram above, the amplitude within the well (A (in)) is larger than the amplitude out of the well (A (out)).

As the probability is related to the square of the amplitude, the probability of finding the Electron out of the Potential well is much less than the probability of finding the Electron within the well, but it is definitely non-zero.

This means that there is a finite probability that the Electron could be found outside of the well, having "tunneled through" the barrier.

The implication of this is that the Electron can be found in a location that classical physics does not allow.

Applications of Quantum Tunneling ( Non-Examinable )

There are several applications of Quantum Tunneling in modern Physics :-

Nuclear Fusion - At the core of a small to medium sized star, the temperature and pressure is not actually sufficient enough to allow fusion, following classical theory. For instance, at the core of the Sun, the temperature is approximately 1x10^{6-7} K, but the Temperature required to overcome the Coulomb Barrier ( Electrostatic repulsion of the nuclei ) allowing fusion is approximately 1x10^{9} K. This means that classically, the Sun cannot undergo nuclear fusion, which obviously is not the case. There is a small probability, however, that the nuclei can penetrate this barrier using Quantum Tunneling. This probability is very small, but the sheer number of nuclei involved means that a massive, sustained nuclear fusion process can occur.

Alpha Decay - For certain radioactive elements classically, Alpha decay should not be possible, yet it is observable. For these isotopes, such as Polonium 212, the Strong Nuclear Force is too large for the Alpha particle to escape the nucleus, however, by Quantum Tunneling, this can be overcome.

Virtual Particle Formation - Quantum Theory allows for the possibility of pairs of particles to spontaneously form within a vacuum, as long as they have an Energy equal to ΔE for a time less than Δt. These particles cannot be measured directly, but their existence as intermediate steps in Radioactive decays or high energy particle collisions means that they can exist.

Hawking Radiation - The formation and subsequent escape by Quantum Tunneling, of a virtual particle is theorised as a method for a way for particle to escape from within a Black Hole. If a pair of virtual particles form at the inside edge of the Event Horizon, one of these particles can pass through the classically impenetrable barrier and leave the Black Hole. As the Energy used to create these particles was "borrowed" from the Black Hole, the escape of a particle causes the Black Hole to lose mass.

Note - The above description of the emission of Hawking Radiation is one of several possible theories of how this radiation is emitted. Also as can, hopefully, be assumed at this stage, this process have been greatly simplified in this instance.