Probability

Please follow these guidelines when writing up your CS 155 homeworks.

A major goal of this class is that you learn how to mathematically analyze probabilistic processes and events. Hence, make sure your proofs are rigor- ous, i.e., that every step is adequately justified. Try to keep your proofs as simple and as concise as possible, keeping in mind that you will potentially have to examine multiple proof strategies in order to achieve this. Make sure to CLEARLY STATE YOUR ASSUMPTIONS at the beginning of any solution, when necessary. As we are doing probabilistic analysis, we shall require that you clearly define the probability space you are working with and identify the events within it that you will analyze.

You are allowed to discuss these problems with other students, but you have to leave any meeting with other students without any written material about the problems, and the writeups must be done by yourself without help from others. If you have any questions regarding how these guidelines apply to a particular problem or what they mean in general, please email the TAs.

The positive square root ∥ψ∥ of ∥ψ∥2 in (3.6) is called the norm of |ψ⟩. As already noted in

Ch. 2, α|ψ⟩ and |ψ⟩ have exactly the same physical significance if α is a non-zero complex number. Consequently, as far as the quantum physicist is concerned, the actual norm, as long as it is positive, is a matter of indifference. By multiplying a non-zero ket by a suitable constant, one can always make its norm equal to 1. This process is called normalizing the ket, and a ket with norm equal to 1 is said to be normalized. Normalizing does not produce a unique result, because eiφ|ψ⟩, where φ is an arbitrary real number or phase, has precisely the same norm as |ψ⟩. Two kets |φ⟩ and |ψ⟩ are said to be orthogonal if ⟨φ|ψ⟩ = 0, which by (3.3) implies that ⟨ψ|φ⟩ = 0.

3.2 Linear Functionals and the Dual Space

Let |ω⟩ be some fixed element of H. Then the function

J 􏰀|ψ⟩􏰁 = I􏰀|ω⟩, |ψ⟩􏰁

assigns to every |ψ⟩ in H a complex number in a linear manner,

J 􏰀α|φ⟩ + β|ψ⟩􏰁 = αJ 􏰀|φ⟩􏰁 + βJ 􏰀|ψ⟩􏰁,

(3.7)

(3.8)

as a consequence of (3.4). Such a function is called a linear functional. There are many different linear functionals of this sort, one for every |ω⟩ in H. In order to distinguish them we could place

3.2. LINEAR FUNCTIONALS AND THE DUAL SPACE 25

a label on J and, for example, write it as J|ω⟩􏰀|ψ⟩􏰁. The notation J|ω⟩ is a bit clumsy, even if its meaning is clear, and Dirac’s ⟨ω|, called a “bra”, provides a simpler way to denote the same object, so that (3.8) takes the form

⟨ω|􏰀α|φ⟩ + β|ψ⟩􏰁 = α⟨ω|φ⟩ + β⟨ω|ψ⟩, (3.9)

if we also use the compact Dirac notation for inner products.

Among the advantages of (3.9) over (3.8) is that the former looks very much like the distributive

law for multiplication if one takes the simple step of replacing ⟨ω| · |ψ⟩ by ⟨ω|ψ⟩. Indeed, a principal virtue of Dirac notation is that many different operations of this general type become “automatic”, allowing one to concentrate on issues of physics without getting overly involved