By far the most widely-used calculational framework in standard QM is the algebraic matrix mechanics approach, pioneered by Heisenberg, Dirac, Hilbert, von Neumann and others in the mid 1920s. It involves state vector representations of the state of a system, encoded via complex-valued vectors representing probability amplitudes (i.e., a Hilbert space). This state vector is a specific way of encoding the configuration space of the entire set of relevant variables, and is thus manifestly non-local, and represents the entire state at a given point in time, in a way that is thus incompatible with the principles of relativity.
This state vector evolves under unitary transformations (rotations in the complex vector space), which preserve the overall magnitudes of the vectors, even as they rotate around in the space. The unitary nature of the rotation transformations represents the behavior of the system when it is being governed by the Schrodinger wave dynamics under the Copenhagen dualistic framework, which perfectly preserves the overall underlying probability space as long as nobody “looks at it the wrong way” (i.e., makes a measurement). Then, at the end, a “measurement” is made by collapsing the probability space down to a single discrete outcome (i.e., along an eigenvector of the resulting state).
This matrix formalism is equivalent to a self-consistent form of probability theory, which can be derived from abstract axioms having nothing to do with quantum physics (Gleason, 1975; Jaynes, 1990; Caves et al., 2002; Fuchs et al., 2014; Mermin, 2018). Indeed, this framework is so general that its only real physical commitment is that quantum physics obeys strict conservation laws, due to the use of unitary state evolution operators.
Thus, the claim that standard QM is such a successful framework must be understood within this context: yes, it is accurate in capturing this basic fact of conservation, but it really isn’t going very far out on a limb here: nothing wagered, nothing lost; but also perhaps not so much gained.
Bra-Ket notation and vector spaces
See bell for a concrete application of all this to the Bell’s inequality test of non-locality.
The basic element of the Hilbert space is an abstract vector, which basically means that it has a dimensionality but, unlike a more concrete vector defined in 3D coordinate space, the basis vectors of the space are arbitrary. The only thing that is defined is a rule for computing the inner product (also known as the dot product), which, notably, is invariant with respect to any arbitrary set of orthogonal, normal-length basis vectors. For real-valued, two-dimensional vectors \(\mathbf{x}\) and \(\mathbf{y}\), the inner product is just the sum of the products of the values:
\[ \langle \mathbf{x}, \mathbf{y} \rangle = x_1 y_1 + x_2 y_2 \]
This inner product defines a distance metric for the space, where the length of a vector is the square root of its inner product with itself.
In quantum applications, the elements of the vectors are complex numbers, and full understanding of these is required before proceeding. In this case, the inner product is defined as multiplication by the complex conjugate, e.g., for \(z = (z_1, z_2) = ({z_1}_r + i{z_1}_i, {z_2}_r + i{z_2}_i)\) where the r and i subscripts denote the real and imaginary components of each value, and likewise for \(w\):
\[ \langle z, w \rangle \equiv z^* w = z^*_1 w_1 + z^*_2 w_2 = {z_1}_r {w_1}_r + {z_1}_i {w_1}_i + {z_2}_r {w_2}_r + {z_2}_i {w_2}_i \]
This is equivalent to the distance you’d get in a 4D space, so the complex values just provide extra “internal” dimensions to the space, in effect.
The bra-ket notation introduced by Paul Dirac provides a compact way of denoting one of these abstract vectors, which typically goes in the “ket” (right-hand side of the word “bra[c]ket”) position, along with an operator that performs some kind of operation on the vector, which is the “bra” and it goes on the left:
\[ \mbox{bra} | \mbox{ket} \rangle \]
(note that the left angle bracket \(\langle\) is omitted here – that is reserved for computing the inner product length measure).
For example, the wave function state \(\psi\) might be a ket vector:
\[ | \psi \rangle \]
The time-evolution of that wave state according to the Schrodinger equation would be indicated by putting this unitary function that transforms the state from one point in time to another, \(U(t,t_0)\) into the bra position:
\[ U(t,t_0) | \psi \rangle \]
We’ll see how this actually works in time evolution below.
You can also just put a constant into the bra position, and you can use arbitrary symbols for the kets:
\[ \frac{1}{\sqrt{2}} | + \rangle + \frac{1}{\sqrt{2}} | - \rangle \]
which indicates a state that is an equal mix of the \(+\) and \(-\) vector states. Typically this notation is used to indicate the spin up (\(+\)) and down (\(-\)) quantum states, where the above expression thus indicates a state that is an even mix or superposition of these two basis states.
Mathematically, \(+\) is typically defined as \((1,0)\) and \(-\) as \((0,1)\) in the spin coordinate space – these are the pure states, which in general you only get after you make a measurement, which, due to the contextual nature of quantum phenomena, rotates whatever starting state you might have into one of these pure eigenstates.
In general, a ket acts like a column vector:
\[ | w \rangle = \begin{pmatrix} w_1 \\ w_2 \end{pmatrix} \]
while a bra acts like a row vector:
\[ \langle z | = (z^*_1, z^*_2) \]
which works out for the dot product notation:
\[ \langle z | w \rangle = (z^*_1, z^*_2) \begin{pmatrix} w_1 \\ w_2 \end{pmatrix} = z^*_1 w_1 + z^*_2 w_2 \]
As you can see, to get this dot product correct for the complex number case, the bra version is subject to the complex conjugate.
You can transform a ket into a bra, which requires taking the conjugate transpose of the ket (also known as a Hermitian transpose):
\[ | z \rangle \rightarrow \langle z | = z^\dagger = \bar{z}^T = (z^*_1, z^*_2) \]
This allows one to construct the tensor product or outer product of two ket states, which is how you construct the configuration space for two otherwise independent particle states:
\[ | a \rangle \otimes | b \rangle = | a \rangle \langle b | = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} (b^*_1, b^*_2) = \begin{pmatrix} a_1 b^*_1 & a_1 b^*_2 \\ a_2 b^*_1 & a_2 b^*_2 \end{pmatrix} \]
Here you can see that the number of elements in the state goes up as an exponential function of the number of times such an outer product is performed, i.e., \(2^N\) for N 2D spin states.
Measurement
A measurement amounts to projecting the quantum state onto a measurement basis state. This is written as:
\[ p(m) = \langle \psi | \Pi_m | \psi \rangle \]
for a measurement m parameterized by the projectors \(\Pi_m\), which is an N x N dimensional matrix that selects a specific subspace of the full N dimensional tensor product space.
The key step in this computation is to compute the density operator \(\rho\), which is the outer product of the quantum state with itself:
\[ \rho = | \psi \rangle \langle \psi | \]
This is effectively the full correlation matrix for every element of the state against every other element. The projector matrix then selects a subspace to project the correlations onto, and the final probability is computed as the trace (sum of the diagonal elements) of the resulting matrix. See below for the full equation, on the more general ensembles case. These diagonal elements of the projection matrix represent the eigenvalues of the projected matrix.
Ensembles
A more general version of the overall framework includes the possibility of multiple different quantum states prepared with different probabilities \(p_j\), in which case the aggregate density matrix can be computed as a weighted function of the probabilities, and then the projection matrix can be applied to that aggregate.
\[ p(m) = tr \left[ \Pi_m \left( \sum_j p_j \rho_j \right) \right] = tr \left[ \Pi_m \left( \sum_j p_j | \psi_j \rangle \langle \psi_j | \right) \right] \]
The critical point here is that the density matrix \(\rho\) contains all the relevant information about all possible measurement outcomes. Therefore, in most cases, that is the single most informative thing to compute. Of course, the actual projections are still needed to predict the outcomes of specific experiments.
Time evolution
The time evolution of a Hilbert space is typically computed using a unitary operator \(U(t)\), which means that it automatically conserves the total vector length of the complex state variables when it is applied. This means that it is effectively a rotation matrix, which in turn means that it can be defined using the natural exponent operator as discussed in rotation by the exponential. Conveniently, the time-independent Schrodinger equation takes the form of this exponential function:
\[ U(t) = e^{-iHt/\hbar} \]
where \(H\) is the Hamiltonian or total energy of the system, which determines the rate of rotation per unit time in the above equation.
As noted above, this is typically used in the following kind of expression:
\[ U(t) | \psi \rangle \]
\[ e^{-iHt/\hbar} | \psi \rangle \]
Thus, the effect of time on the quantum state is to rotate it through the complex plane. This is much simpler than the dynamics of the time-dependent, spatial wave equation version of Schrödinger’s equation.