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Unformatted text preview: m physics — it ﬂies in the face of our intuitions about the physical
world. One way to think about a superposition is that the electron does not
make up its mind about whether it is in the ground state or each of the k − 1
excited states, and the amplitude α0 is a measure of its inclination towards
the ground state. Of course we cannot think of α0 as the probability that
an electron is in the ground state — remember that α0 can be negative or
imaginary. The measurement priniciple, which we will see shortly, will make
this interpretation of α0 more precise. 1.4 The Geometry of Hilbert Space We saw above that the quantum state of the k -state system is described
by a sequence of k complex numbers α0 , . . . , αk−1 ∈ C, normalized so that
j |αj | = 1. So it is natural to write the state of the system as a k dimen- 8 CHAPTER 1. INTRODUCTION sional vector: |ψ = α0
α k −1 The normalization on the complex amplitudes means that the state of the
system is a unit vector in a k dimensional complex vector space — called a
Hilbert space. Figure 1.2: Representation of qubit states as vectors in a Hilbert space. But hold on! Earlier we wrote the quantum state in a very diﬀerent (and
simpler) way as: α0 |0 + α1 |1 + · · · + αk−1 |k − 1. Actually this notation,
called Dirac’s ket notation, is just another way of writing a vector. Thus 1
0 |0 = . , |k − 1 = . .
1 So we have an underlying geometry to the possible states of a quantum
system: the k distinguishable (classical) states |0 , . . . , |k − 1 are represented
by mutually orthogonal unit vectors in a k -dimensional complex vector space.
i.e. they form an orthonormal basis for that space (called the standard basis).
Moreover, given any two states, α0 |0 + α1 |1 + · · · + αk−1 |k − 1, and β |0 +
β |1 + · · β |k − 1, we can compute the inner product of these two vectors,
which is k=0 αj βj . The absolute value of the inner product is the cosine of
the angle between these two vectors in Hilbert space. You should verify that 1.5. BRA-KET NOTATION 9 the inner product of any two basis vectors in the standard basis is 0, showing
that they are orthogonal.
The advantage of the ket notation is that the it labels the basis vectors
explicitly. This is very convenient because the notation expresses both that
the state of the quantum system is a vector, while at the same time explicitly writing out the physical quantity of interest (energy level, position, spin,
polarization, etc). 1.5 Bra-ket Notation In this section we detail the notation that we will use to describe a quantum
state, |ψ . This notation is due to Dirac and, while it takes some time to get
used to, is incredibly convenient. Inner Products
We saw earlier that all of our quantum states live inside a Hilbert space. A
Hilbert space is a special kind of vector space that, in addition to all the usual
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- Fall '13