5 The Incredible Quantum Mechanics
87
This last sentence contains the very core of what is now called Quantum
Mechanics.
What we have said above is truly revolutionary: we can no longer say that
a particle is here or there, but only specify the probability of finding it here or
there. Not only that, if the particle is there, in order to look at it, or find a way
to somehow detect it, we must use a small amount of energy (light waves have
energy) and this energy moves the particle somewhere else. In other words,
the process of observation changes, if ever so slightly, what we have observed.
Gone is the determinism of classical physics, where it was believed that
if one could specify the positions and velocities of all the particles in the
universe, one could predict the future for all eternity, 6 and as an extra bonus
also reconstruct the past. What takes its place is a world where one can predict
nothing with complete certainty, but only estimate the probability that some
event will take place. With macroscopic objects, the uncertainties are negligibly small: if one leaps off a tall building there is very little chance, a second
or two later, of finding oneself reposing in an armchair reading Newton’s Principia. In the domain of atoms, however, analogous phenomena are entirely
possible.
Having now grasped some idea of the relevance of the wave function,
the question arises: how does one calculate it? The answer was provided by
Erwin Schrödinger in 1926, who proposed an equation that bears his name.
The solution of Schrödinger’s Equation for a particle in a particular environment, e.g. an electron in the hydrogen atom, provides the wave function for
that particle. This wave function can be used to calculate essentially everything associated with the particle, i.e. its position, velocity, orbital angular
momentum, etc. However, these quantities are obtained and expressed only
as probabilities.
In our everyday world, we are used to being able to predict events with near
certainty. Astronomers can tell you when solar eclipses will occur hundreds
of years in the future. So what use is it if the best we can get from QM is a
probability? It is like being told that there is a 1 in 36 chance of rolling two
sixes on a pair of dice. It doesn’t help decide whether to bet your hard-earned
cash on the next roll or the one after. (In fact, one shouldn’t waste any time
on this decision. If the dice are “fair”, the chance of rolling two sixes is the
same for every throw, irrespective of past history.)
There are, however, many instances where probabilistic predictions are
rather more useful. Weather forecasts are not exact, but most people consult
them when planning a picnic. Casinos know very accurately the likelihoods of
6 This idea was first put forward by Pierre-Simon Laplace in 1814, and is known as Laplace’s demon
(see Appendix 5.4). It has been debated by physicists ever since.
87
This last sentence contains the very core of what is now called Quantum
Mechanics.
What we have said above is truly revolutionary: we can no longer say that
a particle is here or there, but only specify the probability of finding it here or
there. Not only that, if the particle is there, in order to look at it, or find a way
to somehow detect it, we must use a small amount of energy (light waves have
energy) and this energy moves the particle somewhere else. In other words,
the process of observation changes, if ever so slightly, what we have observed.
Gone is the determinism of classical physics, where it was believed that
if one could specify the positions and velocities of all the particles in the
universe, one could predict the future for all eternity, 6 and as an extra bonus
also reconstruct the past. What takes its place is a world where one can predict
nothing with complete certainty, but only estimate the probability that some
event will take place. With macroscopic objects, the uncertainties are negligibly small: if one leaps off a tall building there is very little chance, a second
or two later, of finding oneself reposing in an armchair reading Newton’s Principia. In the domain of atoms, however, analogous phenomena are entirely
possible.
Having now grasped some idea of the relevance of the wave function,
the question arises: how does one calculate it? The answer was provided by
Erwin Schrödinger in 1926, who proposed an equation that bears his name.
The solution of Schrödinger’s Equation for a particle in a particular environment, e.g. an electron in the hydrogen atom, provides the wave function for
that particle. This wave function can be used to calculate essentially everything associated with the particle, i.e. its position, velocity, orbital angular
momentum, etc. However, these quantities are obtained and expressed only
as probabilities.
In our everyday world, we are used to being able to predict events with near
certainty. Astronomers can tell you when solar eclipses will occur hundreds
of years in the future. So what use is it if the best we can get from QM is a
probability? It is like being told that there is a 1 in 36 chance of rolling two
sixes on a pair of dice. It doesn’t help decide whether to bet your hard-earned
cash on the next roll or the one after. (In fact, one shouldn’t waste any time
on this decision. If the dice are “fair”, the chance of rolling two sixes is the
same for every throw, irrespective of past history.)
There are, however, many instances where probabilistic predictions are
rather more useful. Weather forecasts are not exact, but most people consult
them when planning a picnic. Casinos know very accurately the likelihoods of
6 This idea was first put forward by Pierre-Simon Laplace in 1814, and is known as Laplace’s demon
(see Appendix 5.4). It has been debated by physicists ever since.