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Introduction to quantum computing
Introduction to quantum computing

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3.1 Spin-½ particles

Experiments show that electrons have an intrinsic property which is called spin. (Mass and charge are other examples of intrinsic properties of particles.) Spin is a type of angular momentum with a quantum number of ½ which means that a measurement of spin along an axis can only have values of prefix plus of italic h over two pi solidus two or negative italic h over two pi solidus two as an outcome. (Here italic h over two pi equals h solidus two times pi where h is Planck’s constant, 6.626 × 10-34 J s.) Thus, electrons are referred to as spin-½ particles.

The most important spin operators are the component of spin angular momentum in the z-direction, cap s hat sub z and the total spin angular momentum, cap s hat squared . The eigenvalue equations for these operators are

multiline equation row 1 cap s hat squared vertical line cap s comma cap m sub s mathematical right angle bracket equals cap s times left parenthesis cap s plus one right parenthesis times italic h over two pi squared vertical line cap s comma cap m sub s mathematical right angle bracket comma row 2 cap s hat sub z vertical line cap s comma cap m sub s mathematical right angle bracket equals cap m sub s times italic h over two pi vertical line cap s comma cap m sub s mathematical right angle bracket full stop

where the angled bracket, vertical line mathematical right angle bracket is used and is called a ket. vertical line cap s comma cap m sub s mathematical right angle bracket represents the eigenfunction, known as an eigenstate with a spin quantum number of cap s and a spin magnetic quantum number of cap m sub s . For a single electron cap s equals one divided by two and cap m sub s equals prefix plus minus of one divided by two . The state with cap m sub s equals prefix plus of one divided by two is the spin-up state and the state with cap m sub s equals negative one divided by two is the spin-down state.