- Variation of g(θ), as defined in the text. - Schr¨ odinger’s Cat. - The most famous example is Schr¨ odinger’s “cat paradox” where the cat is in a superposition of the “dead” and “alive” states. - They are extremely fragile, and a very weak coupling to the environment suffices to destroy the quantum superposition of the two states | φ a and | φ b. - 12.1 The Quasi-Classical States of a Harmonic Oscillator. - The energy of the state | n is E n = (n + 1/2)¯ hω.. - 110 12 Schr¨ odinger’s Cat. - (a) Check that if one works with functions of the dimensionless variables X and P , one has. - (c) Using (12.1) for n = 0 and expressing ˆ a in terms of ˆ X and ˆ P, calculate the wave function of the ground state ψ 0 (X) and its Fourier transform ϕ 0 (P. - The Quasi-Classical States. - The eigenstates of the operator ˆ a are called quasi-classical states, for reasons which we now examine.. - Calculate the expectation value of the energy in a quasi-classical state | α . - Following a similar procedure as in question 1.1(c) above, determine the wave function ψ α (X ) of the quasi-classical state | α , and its Fourier trans- form ϕ α (P. - Suppose that at time t = 0, the oscillator is in a quasi-classical state. - (a) Show that at any later time t the oscillator is also in a quasi-classical state which can be written as e − iωt/2 | α(t. - Taking 1.3, and assuming that | α | 1, justify briefly why these states are called “quasi-classical”.. - Assume the state of this pendulum can be described by a quasi-classical state. - 12.2 Construction of a Schr¨ odinger-Cat State. - the Hamiltonian of the system is simply ˆ W . - At time t = 0, the system is in a quasi-classical state. - Show that the states | n are eigenstates of ˆ W , and write the expan- sion of the state | ψ(T ) at time T on the basis. - How does | ψ(T ) simplify in the particular cases T = 2π/g and T = π/g?. - (a) Discuss qualitatively the physical properties of the state (12.3).. - (b) Consider a value of | α | of the same order of magnitude as in 1.6. - In what sense can this state be considered a concrete example of the “Schr¨ odinger cat” type of state mentioned in the introduction?. - 12.3 Quantum Superposition Versus Statistical Mixture. - We now study the properties of the state (12.3) in a “macroscopic” situation. - Consider a quantum system in the state (12.3). - Write the (non-norm- alized) probability distributions for the position and for the momentum of the system. - 12.1 for α = 5i.. - A physicist (Alice) prepares N independent systems all in the state (12.3) and measures the momentum of each of these systems. - For N 1, draw qualitatively the histogram of the results of the N measure- ments.. - 112 12 Schr¨ odinger’s Cat. - Probability distributions for the position and for the momentum of a system in the state (12.3) for α = 5i. - The quantities X and P are the dimensionless variables introduced in the first part of the problem. - The state (12.3) represents the quantum superposition of two states which are macroscopically different, and therefore leads to the paradoxical situations mentioned in the introduction. - Another physicist (Bob) claims that the measurements done by Alice have not been performed on N quantum systems in the state (12.3), but that Alice is actually dealing with a non- paradoxical “statistical mixture”, that is to say that half of the N systems are in the state | α and the other half in the state. - In order to settle the matter, Alice now measures the position of each of N independent systems, all prepared in the state (12.3). - Draw the shape of the resulting distribution of events, assuming that the resolution δx of the measuring apparatus is such that:. - Considering the numerical value obtained in the case of a simple pen- dulum in question 1.6, evaluate the resolution δx which is necessary in order to tell the difference between a set of N systems in the quantum superposition (12.3), and a statistical mixture consisting in N/2 pendulums in the state | α and N/2 pendulums in the state. - 12.4 The Fragility of a Quantum Superposition. - In a realistic physical situation, one must take into account the coupling of the oscillator with its environment, in order to estimate how long one can discriminate between the quantum superposition (12.3) (that is to say the. - “Schr¨ odinger cat” which is “alive and dead”) and a simple statistical mixture (i.e. - If the oscillator is initially in the quasi-classical state | α 0 and if the en- vironment is in a state | χ e (0. - the wave function of the total system is the product of the individual wave functions, and the state vector of the total system can be written as the (tensor) product of the state vectors of the two subsystems:. - The coupling is responsible for the damping of the oscillator’s amplitude. - At a later time t, the state vector of the total system becomes:. - the number α(t) corresponds to the quasi-classical state one would find in the absence of damping (question 1.5(a)) and γ is a real positive number.. - Using the result 1.3, give the expectation value of the energy of the oscillator at time t, and the energy acquired by the environment when 2γt 1.. - For initial states of the “ Schr¨ odinger cat” type for the oscillator, the state vector of the total system is, at t = 0,. - (t) are two normalized states of the environment that are a priori different (but not orthogonal).. - The probability distribution of the oscillator’s position, measured indepen- dently of the state of the environment, is then. - 114 12 Schr¨ odinger’s Cat. - (t) are quasi-classical states. - (a) From the expansion (12.2), show that η = β. - (b) Using the expression found in question 4.1 for the energy of the first os- cillator, determine the typical energy transfer between the two oscillators, above which the difference between a quantum superposition and a sta- tistical mixture becomes unobservable.. - Using the result of the previous question, evaluate the time during which a. - “Schr¨ odinger cat” state can be observed. - 12.5 Solutions. - Section 12.1: The Quasi-Classical States of a Harmonic Oscillator 12.1.1. - We know from the theory of the one-dimensional harmonic oscillator that the energy levels are not degenerated. - The calculation of the norm of | α yields: α | α = e. - The expectation value of the energy is:. - h/2, independently of the value of α.. - 116 12 Schr¨ odinger’s Cat. - The relative uncertainties on the position and on the momentum of the oscil- lator are quite accurately defined at any time. - Hence the name “quasi-classical state”.. - i 3.9 10 9 Section 12.2: Construction of a Schr¨ odinger-Cat State 12.2.1. - If T = 2π/g, then e − ign 2 T = e − 2iπn 2 = 1 and. - (a) For α = iρ, in the state | α , the oscillator has a zero mean position and a positive velocity. - In the state. - The state 12.3 is a quantum superposition of these two situations.. - The state 12.3 is a quantum superposition of such states. - It there- fore constitutes a (peaceful) version of Schr¨ odinger’s cat, where we represent. - Section 12.3: Quantum Superposition Versus Statistical Mixture 12.3.1. - The probability distributions of the position and of the momentum are. - In the latter equation, we have used the fact that, for ρ 1, the two Gaussians centered at ρ. - Alice will find two peaks, each of which contains roughly half of the events, centered respectively at p 0 and − p 0
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