Some Foundations of Quantum Mechanics Johar M. Ashfaque
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Entanglement
The differences between classical and quantum worlds are highlighted most emphatically when we look at the property called entanglement. Entanglement is a property of two or more quantum systems. Here we will consider two systems with H1 and H2 being the associated Hilbert spaces. The Hilbert space of the combined system is then H1 ⊗ H2 . A combined system is said to be entangled if it can not be written in the form |Ψi = |ψ1 i ⊗ |ψ2 i. For example, consider two particles each of which can have one of two states. This is called a qubit. We take a basis of this Hilbert space to be the spin in the z-direction with eigenstates spin up | ↑i or spin down | ↓i. Then the state |Ψi = | ↑i ⊗ | ↓i is not entangled. In contrast, the state 1 |Ψi = √ (| ↑i ⊗ | ↓i − | ↓i ⊗ | ↑i) 2 is entangled. In fact, this is the most famous of all entangled states and is usually known as an EPR pair. Note that this state is a sum over states of the form |Ψi = |ψ1 i ⊗ |ψ2 i and can not be written in a simpler form: this is what makes it entangled. From now on we will simplify our notation by dropping the ⊗ symbol so that the EPR pair is written as 1 |EP Ri = √ (| ↑i| ↓i − | ↓i| ↑i). 2 However, just because a state is written as a sm of terms of the form |Ψi = |ψ1 i ⊗ |ψ2 i does not necessarily mean that it is entangled. Consider for example 1 |Ψi = √ (| ↑i| ↓i + | ↓i| ↑i). 2 This can also be written as |Ψi = | →i| ↓i where | →i =
2
√1 (| 2
↑i + | ↓i) and so this state is not entangled.
The EPR Paradox
In 1935, Einstein, Podolsky and Rosen tried to use the property of entanglement to argue that quantum mechanics is not complete. Ultimately, this attempt failed, revealing instead the jarring differences between quantum mechanics and the classical world.
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Here is the EPR argument. We prepare two particles in the state 1 |EP Ri = √ (| ↑i| ↓i − | ↓i| ↑i) 2 and subsequently separate these particles by a large distance. So Alice and Bob sit in distant locations, each carrying one of the spins of the EPR pair. Let us say that Alice chooses to measure her spin in the z-direction. There are two possibilities: she either finds spin up | ↑i or spin down | ↓i and according to the rules of quantum mechanics, each of these happens with probability 21 . Similarly, Bob can measure the spin of a second particle and also finds spin up or spin down, again with probability 12 . However, the measurements of Alice and Bob are not uncorrelated. If Alice measure the first particle to have spin up then the EPR pair collapses to | ↑i| ↓i which means that Bob must measure the spin of the second particle to have spin down. It would appear, regardless of how far apart they are, the measurement of Alice determines the measurement of Bob. Whatever Alice sees, Bob always sees the opposite. Viewed in the usual framework of quantum mechanics, these correlations arise because of a collapse of the wave-function what happens instantaneously.
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The Review of Spin
We review, here, some of the facts about how we measure the spin along different axes. An operator that measure the spin along the direction a = (sin θ, 0, cos θ) is cos θ sin θ σ · a = σθ = sin θ − cos θ It has eigenvectors θ θ |θ+ i = cos | ↑i + sin | ↓i 2 2
and
θ θ |θ− i = − sin | ↑i + cos | ↓i 2 2
From thism we learn that if we prepare a state in say | ↓i then the probability P (θ± ) of measuring either spin + or spin − along the vector a is P (θ+ ) = sin2
θ 2
and
θ P (θ− ) = cos2 . 2
From the form of the eigenstates |θ± i we that the EPR pair can be written as 1 |EP Ri = √ (|θ+ i|θ− i − |θ− i|θ+ i) 2 for any θ. This means that as long as Alice and Bob both choose to measure the spin along the same direction a then their results will always be perfectly anti-correlated: when one measures spin + the other is guaranteed to measure spin −. This is the special property of the EPR pair that is not shared by other entangled states.
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