Making sense of quantum mechanics/Principles of Quantum Mechanics

What are the first principles of Quantum Mechanics?

All we do is draw little arrows... Richard P. Feynman, 1985

Imagine a bunch of ordinary nails thrown through the air. Each nail will have its own velocity, some will collide, some will cross space unaffected, most will gain spinning motion, each with its own angular velocity, with its own rotation axes. This is just a quantum system. In a quantum system, the elements are represented by arrows, mathematically we call them vectors. A spinning nail has observational properties attached to it (position, translational and angular velocity,...), the same for the vector representing it. The concept vector + observational properties is called a state vector. This is the core of quantum physics.

First principle: A quantum system may be represented by a vector

Top: a pseudo quantum system of needles.
Middle: its representation as vectors (state vectors or kets).
Down: Addition rule for vectors (a set of vectors is just another vector).

Quantum systems known as such are not exactly behaving like the pseudo-quantum system described above, but... there are close similarities. And these similarities give us precious insights in the quantum laws. The first similarity between a "microscopic" quantum system and a bunch of needles, of rods, of arrows or of any linear objects, is the fact that it may be represented by a state vector. This vector bears the observational properties of the system (energy E, position r, momentum p...), which specify the state of the system. Quantum physicists call it a ket, denoted by the marks | > delimiting some symbol(s) informing about the state in which the system is, for example:

$|\psi \rangle \equiv |\psi (E,r,p,...)\rangle \equiv |E,r,p,...\rangle$

which are strictly equivalent notations of a state vector with energy E, position r, momentum p,...

Now imagine we draw a vector on a piece of paper. On another piece of paper, we draw exactly the same vector at the same spot. Both vectors are equal, aren't they? They have same orientation, same position. In ordinary space they are equal. Now let's imagine that the vector on the first piece of paper rotates at 1 turn per second and the vector on the second piece of paper rotates twice as fast. We presently have two different state vectors, the first with angular velocity $\omega$ = 1, the second with angular velocity $\omega$ = 2. Physicists would denote them $|\psi (\omega =1)\rangle$ and $|\psi (\omega =2)\rangle$. So even if the vectors are equal on paper (in ordinary space for 3D drawings), the state vectors they represent are not equal in the quantum-sensical way. Each distinct value of an observational property determines a dimension of the quantum vector state space. In this case, the two values of the angular velocity double the dimensions of ordinary space. Sometimes in the analysis of a physical problem, it is sufficient to discriminate between only two values of a physical property, for example when a rapidly rotating arrow is thrown towards a wire grid. Either it passes (if its rotational plane is closely parallel to the wires) or it is stopped (if its rotational plane has an angle with the wires). In that case, the quantum space is two-dimensional. In other cases there will be 3 possible values, or 4, or more... When there is a continuous set of observational values at stake, the quantum space is infinite dimensional. For example, if different positions determine different states of the vector, ordinary 3D space becomes an infinite dimensional state space.

A set of arrows may be represented by a set of vectors, which is equivalently just another vector (a vector of vectors). If any elementary linear object is represented by a vector, this guarantees that to every system of linear objects also corresponds a vector.

If we keep in mind that state vectors represent ordinary linear objects, there is nothing mysterious about quantum physics. We may deduce some trivial quantum laws, the first of them being that the direction of linear objects varies when physical conditions change.

Second principle: The orientation of the vector representing a quantum system evolves

Quantum evolution laws determine how the orientation of the state vector changes, given some modifications or perturbations of the physical conditions. For example, the time evolution law states that the vector difference of the vectors representing a needle at two infinitesimal close instants $t$ and $t+dt$ is perpendicular to the vector at instant $t$ and proportional to the angular velocity $\omega$ of the needle multiplied by the time difference (see Figure Time evolution of a state vector).

Time evolution of a state vector

An elegant way to represent changes of orientation of vectors is through Argand's method: multiplying by the complex factor $e^{i\theta }$, where $\theta$ is the rotation angle (also called the phase or phase angle). Perpendicularity between two equally normed vectors may therefore be seen as equality between one vector and the other vector multiplied by $e^{i{\pi \over 2}}=i$.

The vector equation pictured in figure "Time evolution of a state vector" therefore writes as:

${d|\psi (\omega ,t)\rangle  \over dt}=-i\omega |\psi (\omega ,t)\rangle$

if the time axis is set in the negative z-direction. This formula is a general deterministic law characterizing the time evolution of systems of linear objects, whether microscopic or macroscopic. For a system of fundamental particles, it usually appears with a factor $\hbar$ at both sides (and i at the left):

$i\hbar {d|\psi (\omega ,t)\rangle  \over dt}=\hbar \omega |\psi (\omega ,t)\rangle$

The factor $\hbar \omega$ multiplying the ket then represents the energy content of the system. This is Schrödinger's time-dependent equation. So we have a second close similarity between quantum systems and macroscopic linear objects: they both obey Schrödinger's equation.

In Schrödinger's time dependent equation, we focus on the way the phase of the vector changes with varying time. If we had focused on another varying parameter (like position, momentum, external energy,...), the formulation of the evolution law would be different, but it would always obey the trivial principle that the orientation of the vector evolves. We will therefore always find a complex factor i in quantum differential equations with terms of odd parity.

The Schrödinger ket equation shows that a ket may be transformed into another ket by means of a differential operation. In this differentiation, a proportionality factor $\hbar \omega$ emerges which has some physical meaning. This leads us to our third quantum principle.

Third principle: Kets are transformed into other kets by means of operations that reveal an observational property

In the time dependent Schrödinger equation, the operator $i\hbar {d \over dt}$ operates on the ket $|\psi \rangle$ giving the same ket multiplied by the factor $\hbar \omega$. For elementary particles, this factor is the measure of the energy of the particle.

The general form of the equation where a ket (just an arrow) is transformed into another ket, shows as:

${\hat {A}}|\psi \rangle =|\chi \rangle$

where $|\psi \rangle$ and $|\chi \rangle$ may be any imaginable ket and ${\hat {A}}$ denotes the appropriate operator. In quantum physics, operators are denoted by a circumflex on the letter.

The operator $i\hbar {d \over dt}$ of the Schrödinger equation is called the Hamiltonian operator ${\hat {H}}$. Operating with the Hamiltonian on a ket extracts the energy from the ket.

In the preceding section, we introduced Argand's construction because it allowed us to formulate the rotation of a vector through an angle $\phi$ in a very powerful way: multiplying the vector by $e^{i\phi }$. Through the rotation operator ${\hat {R}}(\phi )$, the ket $|\psi \rangle$ is transformed into a rotated ket and we may write:

${\hat {R}}(\phi )|\psi \rangle =|\chi \rangle =e^{i\phi }|\psi \rangle$.

Another example of an operator is the time evolution operator Û(t, t+dt) (letting time pass by from instant t to t+dt). For a freely rotating vector with constant angular speed $\omega$, Û(t, t+dt) operating over $|\psi \rangle$ yields the equation:

${\hat {U}}(t,t+dt)|\psi (t)\rangle =e^{-i\omega dt}|\psi (t)\rangle =|\psi (t+dt)\rangle$.

Û is a unitary operation because it leaves the length of the ket unchanged. It has physical meaning because operating with it on a ket representing the state of a linear object results in another ket representing also a physical state (its state at a subsequent instant).

In the preceding equation, Û acts on the ket $|\psi (t)\rangle$ and gives a resultant ket $|\psi (t+dt)\rangle$ at a subsequent instant. We could also see it the other way round: what is the operator that would act on the ket $|\psi (t+dt)\rangle$ and give $|\psi (t)\rangle$? That operator is the reversed operator of Û, ${\hat {U}}^{-1}$, equivalent to the conjugate operator Û* in case of unitary operators, where the phase is changed by $\omega dt$:

${\hat {U}}^{-1}(t,t+dt)|\psi (t+dt)\rangle =e^{i\omega dt}|\psi (t+dt)\rangle =|\psi (t)\rangle$.

Example: Particle in a box

The four lowest energy states of a rotating arrow bouncing back and forth in a box

The quantum principles of preceding sections may be applied to the behavior of an arrow in a one-dimensional box of length L. Let the arrow bounce back and forth with constant velocity $v_{c}$ between two parallel walls. The period between two bounces is $L/v_{c}$. The needle is plane rotating in the x-z plane with angular velocity $\omega$. We will assume that the arrow acquires rapidly a stationary state characterized by the fact that it is parallel to the wall at each instant of bouncing. Therefore, at equilibrium, the arrow must rotate about an angle $n\ \pi$ between two bounces, with n integer-valued. This condition is expressed as:

$\omega =n\ \pi \ v_{c}/L$.

The rotational state of the arrow is quantized due to the boudary conditions. n may take any integer value between $-\infty$ and $+\infty$.

We may characterize the state of the arrow by the ket $|\psi _{0}(\omega _{n})\rangle$, say a representation of the needle with angular velocity $\omega _{n}$ at the instant of bouncing at the left wall (taken as t=0). We also set x=0 at that wall. At a later timestamp, the state of the needle is represented by the initial ket, multiplied by its phase factor:

$|\psi (t,\omega _{n})\rangle =exp(-i\omega _{n}t)|\psi _{0}(\omega _{n})\rangle$.

Once the in initial ket given, we may as well characterize the state of the needle by the complex scalar function $exp(-i\omega _{n}t)$. This function is called the wave-function of the needle and is generally denoted by the ket without the brackets: $\psi (t,\omega _{n})$. We may write:

$|\psi (t,\omega _{n})\rangle =\psi (t,\omega _{n})|\psi _{0}(\omega _{n})\rangle$,

or more concisely:

$|\psi \rangle =\psi \ .\ |\psi _{0}\rangle$

holding in mind that we are dealing with functions of time.

Superposition of possible states of the rotating arrow in a box depending on the x-coordinate

We could however see it otherwise, forgetting the time parameter and concentrating on the x-coordinate of the needle. For this stationary state, the phase of the arrow is also a function of the x-coordinate. We may therefore as well write the preceding equation as:

$|\psi (x)\rangle =exp(-ikx)|\psi _{0}\rangle$,

with k the wave-number $n\pi L$ and $|\psi _{0}\rangle$ the ket at x=0. Or concentrating only on the direction of the needle, we could write:

$|\psi (\phi )\rangle =exp(-i\phi )|\psi _{0}\rangle$.

These equations are equivalent for this particular stationary state in a 1-dimensional box. We could write them in different ways, even involving other physical properties such as the direction of the rotation axis, the energy, the momentum, the potential or any imaginable physical property of the system. They illustrate that, given an initial ket, the real issue of quantum mechanics is the complex (wave-)function that operates on the ket. That function contains all that can be known about the physics of the system. It tells us how a ket is transformed into another ket, it tells us which physical observables can be extracted from it and how they are related.

When we look at the figure of that arrow that bounces back and forth in a box, we may grasp it as a whole, visualizing the motion classically, knowing how to describe the configuration deterministically. However, at the quantum level, observations are discrete. For example, when we observe the position of the arrow quantum-mechanically, we in fact only notice the location of the interaction of that arrow with another quantum particle (arrow). That location is a point, while the arrow is extended. So we have an intrinsic indeterminacy in the measurement of the position. The same for the direction of the arrow. Because measurements at the quantum level are discrete, it is impossible to determine the phase of the arrow through a single observation. There is an intrinsic indeterminacy of an angle $2\pi$ in the phase $\phi$ of the quantum particle. This leads us to our fourth quantum principle, the Heisenberg indeterminacy principle.

Fourth principle: In quantum measurements, the result is always undetermined

Physically, quantum measurements rest on the observation of interactions between two quantum particles. An interaction between two arrows is located at a point, while the arrows are extended. A measurement outcome therefore only shows us a facet of the state of the arrow.

If we try to measure the position of the arrow, there is an indeterminacy in the outcome that equals (twice) the length L of the arrow. For the same event, the arrow could as well have been observed at any place between x - L and x + L.

The phase of the arrow (its direction projected on a plane) is totally undetermined. If we denote the indeterminacy of the phase as $\Delta \phi$, we may write:

$\Delta \phi =\Delta k\ .\ \Delta x=\Delta \omega \ .\ \Delta t=2\pi$.

Because measurement outcomes are undetermined, quantum mechanics only give us statistical means to make predictions. This leads us to the next principle: the probability of an experimental outcome is deduced from the probability that the observed arrow interacts with the observing arrow.

Fifth principle: Quantum probabilities involve interaction cross sections of both observed and observing particles

Imagine we want to observe quantum-mechanically the arrow in the particle box of the figure. In order to 'detect' that arrow, we need a second arrow (that will also bounce back and forth between both walls). The probability of interaction between both arrows is therefore proportional to the projection of the first arrow multiplied by the projection of the second arrow perpendicularly to the line of interaction, which is just the wavefunction squared. This principle is known as Born's rule