Ket Space

Consider a microscopic system composed of particles, or microscopic bodies, with specific properties (mass, moment of inertia, etc.) interacting according to specific laws of force. There will be various possible motions of the particles, or bodies, consistent with these laws of force. Let us term each such motion a ``state'' of the system. According to the principle of superposition of states, any given state can be regarded as a superposition of two or more other states. Thus, states must be related to mathematical quantities of a kind that can be added together to give other quantities of the same kind. The most obvious examples of such quantities are vectors.

Let us consider a particular microscopic system in a particular state, which we label $A$ . For example, a photon with a particular energy, momentum, and polarization. We can represent this state as a particular vector, which we also label $A$ , residing in some vector space, where the other elements of the space represent all of the other possible states of the system. Such a space is called a ket space (after Dirac), and the vectors that reside in this space are known as kets [32]. The ket that corresponds to state $A$ is conventionally written

$$\vert A\rangle.$$ (1.1)

Suppose that state $A$ is, in fact, the superposition of two different states, $B$ and $C$ . This inter-relation is represented in ket space by writing

$$\vert A\rangle = \vert B\rangle + \vert C\rangle,$$ (1.2)

where $\vert B\rangle$ is the ket relating to the state $B$ , et cetera. For instance, state $B$ might represent a photon propagating in the $z$ -direction, and plane polarized in the $x$ -direction, and state $C$ might represent a similar photon plane polarized in the $y$ -direction. In this case, the sum of these two states represents a photon whose direction of polarization makes an angle of $45^\circ$ with both the $x$ - and $y$ -directions (by analogy with classical physics) [64]. This latter state is represented by the ket $\vert B\rangle+\vert C\rangle$ .

Suppose that we wish to construct a state whose direction of polarization subtends an arbitrary angle $\alpha$ with the $x$ -direction. We can do this via a suitably weighted superposition of states $B$ and $C$ . By analogy with classical physics, we require $\cos\alpha$ of state $B$ , and $\sin\alpha$ of state $C$ [64]. This new state is represented by the ket

$$\cos\alpha\,\vert B\rangle + \sin\alpha\,\vert C\rangle.$$ (1.3)

Note that we cannot form a new state by superposing a state with itself. For instance, a photon polarized in the $y$ -direction superposed with another photon polarized in the $y$ -direction (with the same energy and momentum) gives the same photon. This implies that the ket

$$c_1\, \vert A\rangle + c_2\, \vert A\rangle = (c_1 + c_2)\, \vert A\rangle$$ (1.4)

corresponds to the same state that $\vert A\rangle$ does. Here, $c_1$ and $c_2$ are real numbers. Thus, kets differ from conventional vectors in that their magnitudes, or lengths, are physically irrelevant. All of the possible states of the system are in one to one correspondence with all of the possible directions of vectors in the ket space, no distinction being made between the directions of the kets $\vert A\rangle$ and $-\vert A\rangle$ . There is, however, one caveat to the previous statements. If $c_1+c_2 = 0$ then the superposition process yields nothing at all. In other words, no state. The absence of a state is represented by the null ket, $\vert\rangle$ . The null ket has the fairly obvious property that

$$\vert A\rangle + \vert\rangle = \vert A\rangle$$ (1.5)

for any ket $\vert A\rangle$ . The reason that kets pointing in the same direction represent the same state is ultimately related to the quantization of matter. In other words, the fact that matter comes in irreducible packets called photons, electrons, atoms, et cetera. If we observe a microscopic system then we either see a state (i.e., a photon, an atom, or a molecule, etc.) or we see nothing--we can never see a fraction, or a multiple, of a state. In classical physics, on the other hand, if we observe a wave then the amplitude of the wave can take any value between zero and infinity. Thus, if we were to represent a classical wave by a vector then the magnitude, or length, of the vector would correspond to the amplitude of the wave, and the direction would correspond to the frequency and wavelength, so that two vectors of different lengths pointing in the same direction would represent different wave states.

We have seen, in Equation (1.3), that any plane polarized state of a photon can be represented as a linear superposition of two orthogonal polarization states in which the weights are real numbers. Suppose, however, that we wish to construct a circularly polarized photon state. We know from classical physics that a circularly polarized wave is a superposition of two waves of equal amplitude, plane polarized in orthogonal directions, that oscillate in phase quadrature [64]. This suggests that a circularly polarized photon is the superposition of a photon polarized in the $x$ -direction (state $B$ ) and a photon polarized in the $y$ -direction (state $C$ ), with equal weights given to the two states, but with the proviso that state $C$ oscillates $90^\circ$ out of phase with state $B$ . By analogy with classical physics, we can use complex numbers to simultaneously represent the weighting and relative phase in a linear superposition [51]. Thus, a circularly polarized photon is represented by the ket

$$\vert B\rangle + {\rm i}\, \vert C\rangle.$$ (1.6)

Likewise, a general elliptically polarized photon is represented by

$$c_1\, \vert B\rangle + c_2\, \vert C\rangle,$$ (1.7)

where $c_1$ and $c_2$ are complex numbers [51]. We conclude that a ket space must be a complex vector space if it is to properly represent the mutual inter-relations between the possible states of a microscopic system.

Suppose that the ket $\vert R\rangle$ is expressible linearly in terms of the kets $\vert A\rangle$ and $\vert B\rangle$ , so that

$$\vert R\rangle = c_1\, \vert A\rangle + c_2 \,\vert B\rangle.$$ (1.8)

We say that $\vert R\rangle$ is dependent on $\vert A\rangle$ and $\vert B\rangle$ . It follows that the state $R$ can be regarded as a linear superposition of the states $A$ and $B$ . So, we can also say that state $R$ is dependent on states $A$ and $B$ . In fact, any ket (or state) that is expressible linearly in terms of certain others is said to be dependent on them. Likewise, a set of ket (or states) are termed independent if none of them are expressible linearly in terms of the others.

The dimensionality of a conventional vector space is defined as the number of independent vectors contained in that space. Likewise, the dimensionality of a ket space is equal to the number of independent kets that it contains. Thus, the ket space that represents the possible polarization states of a photon propagating in the $z$ -direction is two dimensional (the two independent kets correspond to photons plane polarized in the $x$ - and $y$ -directions, respectively). Some microscopic systems have a finite number of independent states (e.g., the spin states of an electron in a magnetic field). If there are $N$ independent states then the possible states of the system are represented as an $N$ -dimensional ket space. Some microscopic systems have a denumerably infinite number of independent states (e.g., a particle trapped in an infinitely deep, one-dimensional, potential well). The possible states of such a system are represented as a ket space whose dimensions are denumerably infinite. Such a space can be treated in more or less the same manner as a finite-dimensional space. Unfortunately, some microscopic systems have a non-denumerably infinite number of independent states (e.g., a free particle). The possible states of such a system are represented as a ket space whose dimensions are non-denumerably infinite. This type of space requires a slightly different treatment to spaces of finite, or denumerably infinite, dimensions. (See Section 1.15.)

In conclusion, the states of a general microscopic system can be represented as a complex vector space of (possibly) infinite dimensions. Such a space is termed a Hilbert space by mathematicians.