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Unformatted text preview: nce of intuition for
quantum entities. Why should they be intuitive? Things on our scale do
not behave like wavefunctions, and unless we conduct wild experiments like
this we do not see the eﬀects of quantum mechanics. The following sections
describe in more detail some of the basic truths of quantum mechanics, so
that we can build an intuition for a new behavior of matter. 1.2 The Axioms of Quantum Mechanics “I think I can safely say that nobody understands quantum mechanics.”
Paradoxically, the fundamental principles of quantum mechanics can be
stated very concisely and simply. The challenge lies in understanding and
applying these principles, which is the goal of the rest of the book. Here are
• The superposition principle explains how a particle can be superimposed
between two states at the same time.
• The measurement principle tells us how measuring a particle changes its
state, and how much information we can access from a particle.
• The unitary evolution axoim governs how the state of the quantum system evolves in time.
In keeping with our philosophy, we will introduce the basic axioms gradually, starting with simple ﬁnite systems, and simpliﬁed basis state measurements, and building our way up to the more general formulations. This should
allow the reader a chance to develop some intuition about these topics. 1.3 The Superposition Principle Consider a system with k distinguishable (classical) states. For example, the
electron in a hydrogen atom is only allowed to be in one of a discrete set of
energy levels, starting with the ground state, the ﬁrst excited state, the second
excited state, and so on. If we assume a suitable upper bound on the total 1.4. THE GEOMETRY OF HILBERT SPACE 7 energy, then the electron is restricted to being in one of k diﬀerent energy
levels — the ground state or one of k − 1 excited states. As a classical system,
we might use the state of this system to store a number between 0 and k − 1.
The superposition principle says that if a quantum system can be in one of
two states then it can also be placed in a linear superposition of these states
with complex coeﬃcients.
Let us introduce some notation. We denote the ground state of our k -state
system by |0, and the succesive excited states by |1 , . . . , |k − 1. These are
the k possible classical states of the electron. The superposition principle tells
us that, in general, the quantum state of the electron is α0 |0 + α1 |1 + · · · +
−1 |k 2 1, where α0 , α1 , . . . , αk−1 are complex numbers normalized so that
|αj | = 1. αj is called the amplitude of the state |j . For instance, if k = 3,
the state of the electron could be
|ψ = √ |0 + |1 + |2
|ψ = √ |0 − |1 + |2
|ψ = 1+i
1 + 2i
| 0 −
| 1 +
| 2 .
3 The superposition principle is one of the most mysterious aspects about
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This document was uploaded on 09/22/2013.
- Fall '13