Category: 3.The Essentials of Quantum Mechanics

Definition 3.1. The tensor product of two matrices A of dimensions m × n and B of any dimensions is given by A ⊗ B =       A 11 B A 12 B . . . A 1n B A 21 B A 22 B . . . A 2n…

We would in general consider not just a single quantum system, represent- ing one qubit, but a multiple qubit system that will consist of distinct and non-interacting component single-qubit systems. The quantum states of the composite system are elements of a larger Hilbert space composed of the single qubit Hilbert spaces. For this we take…

Someone gives you an electron and asks you: what is the spin? How will you answer? If you measure S x , S y , or S z you will get one of two answers, at random. Any observable you measure gives one of its eigenvalues at random. The state has probabilistic information about each…

From the components of two vectors, we can construct a matrix by the outer product. For vectors |v 1 i = [a 1 a 2 …a n ] T and |v 2 i = [b 1 b 2 …b n ] T , this is denoted by |v 1 ihv 2 | and represented by…

An operator is said to be self-adjoint if it satisfies hv| ˆ A|wi = hv| ˆ A † |wi. (3.7) The corresponding matrix is said to be Hermitian. An important conse- quence of self-adjointness is that the eigenvalues will turn out to be real. A self-adjoint operator is thus a good candidate for a physical…

The state space may be said to be defined by its basis states. How do we identify the basis? We have said that when we measure a physical quantity, the state corresponding to the value measured is a basis state for the sys- tem. This brings us directly to the question: which physical quantity shall…

Inner product of vectors gives the component of one vector in the direction of the other. Similarly for quantum states, the inner product hψ|φi is the probability amplitude that one state is along the other. For example, |ψi = 1 √ 3 |0i + r 2 3 |1i. Then the inner product h0|ψi = 1…

This is the same as the condition hψ|ψi = 1, i.e., the state |ψi must be normalized, or be a unit vector in Hilbert space. A state vector that does not have unit norm can be normalized by dividing it by its norm. The first axiom of quantum mechanics is a statement that embodies all…

In order to be able to define orthogonality and the “size” of a vector, we need the notion of an inner product. This is just like the dot product of two vectors. This is basically a rule for assigning a (complex) number to a pair of vectors. For this we define a dual vector space…

We saw in the previous chapter how to describe the spin state of an elec- tron. 2 The “system” in this case is just that property of an electron that responds to a gradient in an applied magnetic field. The state of this system is a member of a 2-dimensional vector space. This is because…