Quantum superposition is a fundamental principle of quantum mechanics that states that linear combinations of solutions to the Schrödinger equation are also solutions of the Schrödinger equation.
denote particular solutions to the Schrödinger equation in Dirac notation weighted by the two probability amplitudes
The interference fringes in the double-slit experiment provide another example of the superposition principle.
Furthermore, this differential equation is restricted to be linear and homogeneous.
, a linear combination of those solutions also solve the wave equation:
The quantum wave equation can be solved using functions of position,
This transformation is itself a quantum superposition and every position wave function can be represented as a superposition of momentum wave functions and vice versa.
These superpositions involve an infinite number of component waves.
Important mathematical operations on quantum system solutions can be performed using only the coefficients of the superposition, suppressing the details of the superposed functions.
This leads to quantum systems expressed in the Dirac bra-ket notation:[1]: 245
This approach is especially effect for systems like quantum spin with no classical coordinate analog.
Such shorthand notation is very common in textbooks and papers on quantum mechanics and superposition of basis states is a fundamental tool in quantum mechanics.
Paul Dirac described the superposition principle as follows: The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, A and B, such that there exists an observation which, when made on the system in state A, is certain to lead to one particular result, a say, and when made on the system in state B is certain to lead to some different result, b say.
What will be the result of the observation when made on the system in the superposed state?
The answer is that the result will be sometimes a and sometimes b, according to a probability law depending on the relative weights of A and B in the superposition process.
The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states.
[2]Anton Zeilinger, referring to the prototypical example of the double-slit experiment, has elaborated regarding the creation and destruction of quantum superposition: "[T]he superposition of amplitudes ... is only valid if there is no way to know, even in principle, which path the particle took.
It is important to realize that this does not imply that an observer actually takes note of what happens.
It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘‘out there.’’ The absence of any such information is the essential criterion for quantum interference to appear.
[3]Any quantum state can be expanded as a sum or superposition of the eigenstates of an Hermitian operator, like the Hamiltonian, because the eigenstates form a complete basis: where
basis and is called the wave function of the particle.
can be expanded as a superposition of an infinite number of basis states.
indexes the set of eigenstates of the Hamiltonian with energy eigenvalues
we see immediately that where is a solution of the Schrödinger equation but is not generally an eigenstate because
As previously discussed, the magnitudes of the complex coefficients give the probability of finding the electron in either definite spin state: where the probability of finding the particle with either spin up or down is normalized to 1.
is the sum of the tensor products of the position space wave functions and spinors.
Successful experiments involving superpositions of relatively large (by the standards of quantum physics) objects have been performed.
[11]: 13 Unlike classical bits, a superposition of qubits represents information about two states in parallel.
[11]: 31 Controlling the superposition of qubits is a central challenge in quantum computation.
Qubit systems like nuclear spins with small coupling strength are robust to outside disturbances but the same small coupling makes it difficult to readout results.