No paradox just an absence of intuition for quantum

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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.” -Richard Feynman 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 two basic • 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￿ + · · · + αk − ￿−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, j the state of the electron could be 1 1 1 |ψ ￿ = √ |0￿ + |1￿ + |2￿ 2 2 2 or 1 1 i |ψ ￿ = √ |0￿ − |1￿ + |2￿ 2 2 2 or |ψ ￿ = 1+i 1−i 1 + 2i | 0￿ − | 1￿ + | 2￿ . 3 3 3 The superposition principle is one of the most mysterious aspects about quantu...
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This document was uploaded on 09/22/2013.

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