Author: workhouse123
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Inner product
In order to be able to define orthogonality and the “size” of a vector, we need the notion of an inner product. This is just like the dot product of two vectors. This is basically a rule for assigning a (complex) number to a pair of vectors. For this we define a dual vector space…
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Basis states
We saw in the previous chapter how to describe the spin state of an elec- tron. 2 The “system” in this case is just that property of an electron that responds to a gradient in an applied magnetic field. The state of this system is a member of a 2-dimensional vector space. This is because…
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The State Space
When you describe the state of a physical system, you collect all the pa- rameters required to fully specify it: for instance, the state of a ball may be specified by its position in space, its velocity, and maybe its rate of spin; the state of a volume of gas by its temperature and its…
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Introduction
How do we describe the state of a quantum mechanical physical system in a mathematically precise way? How do we ascribe physical properties to this system? How does the system evolve in time? How do its properties change when it interacts with another system or a force? How do we measure and determine its properties?…
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Interference in the Stern–Gerlach setup
Beam Amplitudes Spot intensity on screen Left Right as SG x is slowly turned off L R L R Final (SG x off) + + + − no spot − + no spot − − In the language of optics, we can say that the middle two beams, having 26 Introduction to Quantum Physics and…
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Interference and Complex Amplitudes
Consider a sequence of Stern–Gerlach tests where an unpolarized beam passes through an SG z and subsequently through an SG x . Clearly there are four possible outcomes: a screen placed at the end will show up 4 spots as in Figure 2.8. If we place another SG z at the end, we expect eight…
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An Experiment to Illustrate Superpositions
An experiment such as that in Figure 2.6 represents a measurement out- come: the average value of the z-component of spin of the input beam is +1/2, obtained by the weighted probabilities of spin 1/2 at the up port and spin −1/2 at the down port, which in quantum notation is: hS z i |ii…
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Choice of different bases
The magnetic moment, and therefore spin, has three spatial components, and we think of spin as a vector in space with x, y, and z components. To completely determine the spin of an electron, we would need to measure all three components. We could do this by setting up SG machines with magnetic inhomogeneities along…
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Superpositions
A generic (unknown) state |ψi, before measurement in a particular basis, has the potential to be in either basis state. Suppose the probability am- plitudes for measuring the state to be |↑i is a complex number α and that for |↓i is β. We express this fact mathematically by writing |ψi as a linear superposition…
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Introduction to Quantum Physics and Information Processing
Let’s denote the (unknown but pure) input state of the electrons in the beam by |ii and the output state by |oi. The probability of obtaining this particular output is given by P(|ii → |oi) = no. of electrons in state |oi Total number of electrons in input beam . (2.2) We are going to…