Question

Need the answer for a)

1. The idea of a quantum fluctuation is central to much of physics, including cosmology; for example, it is believed that the current structure of the Universe arose from small initial quantum fluctuations that were stretched out during a period of rapid Universal expansion called inflation. As we will discuss later, it turns out that we can represent quantum systems as an infinite collection of harmonic oscillators. For each of the harmonic oscillators, the amplitude of fluctuations, δx is defined as δx ≡ p h|x| 2i. (a) Recall that the raising and lowering operators for a quantum simple harmonic oscillator are defined as aˆ± ≡ 1 √ 2m~ω (∓ipˆ+ mωxˆ). Show that (ˆa±) ∗ = ˆa∓, and determine ˆx in terms of ˆa±. (b) Noting that, for a quantum simple harmonic oscillator state |ni, we have ˆa+|ni = √ n + 1|n + 1i, and ˆa−|ni = √ n|n − 1i, then determine δxn for a given state |ni (note that this is really just the uncertainty ∆x). (c) Using your results from part (b), what is the amplitude of fluctuations for the ground (vacuum) state |0i?

1. The idea of a quantum fluctuation is central to much of physics, including cosmology; for example, it is believed that the current structure of the Universe arose from small initial quantum fluctuations that were stretched out during a period of rapid Universal expansion called inflation. As we will discuss later, it turns out that we can represent quantum systems as an infinite collection of harmonic oscillators. For each of the harmonic oscillators, the amplitude of fluctuations, Ox is defined as 8x = (1212). (a) Recall that the raising and lowering operators for a quantum simple harmonic oscillator are defined as ât = Tone (Fip + mwî). V2mħw Show that (â+)* = âF, and determine î in terms of â+. (b) Noting that, for a quantum simple harmonic oscillator state [n), we have â+|n) = Vn+1 n + 1), and â_n) = nn – 1), then determine dxn for a given state in) (note that this is really just the uncertainty Ar). (c) Using your results from part (b), what is the amplitude of fluctuations for the ground (vacuum) state (0)?

Answer 1

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