The Heisenberg uncertainty principle

Introduction

The principle states that:

It’s not possible to predict perfectly both position and momentum of a moving object.

This principle doesn’t change the description of macroscopic-objects motion: for instance we can talk about the trajectory of a car negotiating a curve, or about the trajectory of a planets around the sun. Both these objects have very definite position and velocity at each instant in space (at least in the common sense) and, as a matter of fact, their motion can be described successfully by using the rules of Newton’s classical physics.

On the contrary the Heisenberg uncertainty principle has a critic role in the description of micro-objects like atom, electrons, an so on…In the cases of micro-objects it’s really impossible to predict perfectly well both position and Momentum at a given instant.

For this reason the most successful description of the behaviour of micro objects behaviour is not the classical one but the Shrödinger’s one (Quantum Mechanics): the wave-like description avoid the problem of defining trajectories and velocity for electrons inside atoms, but at the same time permits an accurate prediction for the behaviour of these atoms.

Getting an insight into Heisenberg’s principle.

Heisenberg gave a mathematical demonstration of the principle of which final statement is usually written in the following way:

( is Planck’s constant)

What are and ?

For an object with a mass and velocity , we can define the Momentum of the object as

We can also indicate the position of the object in space with the letter .

Hence represents the uncertainty in the position of the object, the error in measuring the position of the object,

while represents the uncertainty in the Momentum of the object, the error committed in measuring the Momentum.

means that the product of the uncertainties of position and Momentum can’t decrease under the value of Planck’s constant . In other words it’s not possible to know both quantities perfectly: if the uncertainty in position is small, the uncertainty in Momentum must be large and vice versa. At most, we can know exactly the position but nothing about the Momentum (almost no error on but a very large error on ) and vice versa.

The Heisenberg’s principle is a fundamental limitation on what we can observe rather than a lack of experimental capability. It derives directly from nature in itself as we’re going to explain with a plausible line of reasoning.

In order to understand the principle here are some thoughts on the following arguments:

First argument.

To make a measurement on one object is equivalent to establish an interaction between two objects:

1.  the observed object.
2.  the instrument we use to make the measurement on the observed object (the observing subject).

Second argument.

Imagine the interaction between a wave W, playing the role of the observing subject, and the object O (observed object): the position of the object will be pointed out by the wave only if its wavelength is commensurable with the dimension of the object. If not, the wave doesn’t reveal the existence of the object. For instance you can think of a small boat on the sea:

In situation A (wavelength longer than the ship dimension), the wave goes over the boat, the wave is not disturbed by the presence of the boat, the wave doesn’t change its shape and its direction because of the boat, the wave continues to move in the same way as if the boat doesn’t exist.

If we use such a wave to localize the boat we deduce there’s no boat because of the low interaction of the wave with the boat.

In the situation B (wavelength commensurable or shorter than the ship dimension), the wave doesn’t go over the ship: the wave is reflected by the boat and comes back. This wave is more useful than the previous to localize the boat; if we think of this wave as a signal that reveals the position of the boat, we can say that this signal strongly interacts with the ship, it’s reflected and comes back, and from the backing signal we can deduce the existence of the boat and its position.

Thinking of one electron inside an atom we can do a comparison with the previous example.

The electron represents the boat, the waves we use to reveal its positions are electromagnetic waves (light). We "light up" the electron in order to localize it. The quality of information about the electron position is strongly related to the wavelength of the light we use, only if the wavelength of the light is commensurable with the dimension of the electron do we have a precise information about its position.

Third argument.

In measuring the Momentum

we are involved in the problem of a velocity measurement. From the conceptual point of view, thinking of an object while it is moving, an instantaneous velocity measurement always implies the measurement of the position of the object at two very close but distinct points in space: velocity is the displacement between these two points over the time elapsed to cover that distance.

Fourth argument.

During an interaction between two objects A and B, the greater the energy of the object A, the higher the perturbation introduced in the state of the object B. For instance we can think of two balls on a billiard table: the greater the kinetic energy of the first ball the higher the change in Momentum of the second ball when struck by the first ball. As a matter of fact because of the interaction, the second ball experiences a change in its state (change in speed and direction, hence change in Momentum) directly connected to the energy of the first ball.

If we apply the previous arguments to one electron inside an atom, we conclude that is not possible to establish its position and velocity perfectly well at a given instant:

In order to localize it well, we have to use light with very short wavelength (second argument). Light with a very short wavelength gets big frequency ( and are inversely proportional), but a photon of light with a high frequency is very energetic (Energy is proportional to the frequency) and for this reason strongly perturbs the electron we want to observe (fourth argument). In particular such a photon strongly perturbs the momentum of the electron causing its direction and speed to change like a gun bullet does when shot against a billiard ball. This fact makes the electron Momentum measurement impossible (third argument) because the interaction produce a strong alteration of the velocity of the electron.

So, to localize the electron perfectly well in the atom, we lose all the information about its momentum, because we destroy this information during the interaction.

We have to come to a compromise: Using light of higher wavelength we lost some information about the position of the electron (we have uncertainty on the position), but light with higher wavelength gets shorter frequency and it is less energetic, for this reason the electron is less perturbed and lets us to deduce some information on the momentum (even if with a given uncertainty).

This is the essence of the principle: if we improve the information on one quantity we lose information on the quantity conjugated, the uncertainties on the two measurements are inversely connected and the principle states that they can’t decrease under the value of the Planck’s constant . Using maths:

.

This is a plausible line of reasoning in order to understand the principle, as a matter fact, to observe inside an atom (whose linear dimension is of the order of ) we should use e.m. waves like (gamma) rays, but these rays are so energetic that if we use them on the atom we ionised it!