Reconciling quantum with classical mechanics

[physikerin]

Homework Statement

The pendulum of a grandfather clock has a period of 1 s and makes excursions of 3 cm either side of dead centre. Given that the bob weighs 0.2 kg, around what value of n would you expect its non-negligible quantum amplitudes to cluster?

The Attempt at a Solution

I am very uncertain about this attempt. I know that a measurement of energy is guaranteed to yield a value that lies close to E=E_n where I think that E_n is given by E_n=(n + 1/2) hbarw. So I thought that I could calculate the potential energy at the highest point of the oscillation (by using the fact that T=2pi(L/g)^1/2 and working out how high the bob swings) and equate that to E_n and then work out n. The answer I got for that however was phenomenally large however (around 3.63*10 to the 32 or so) and while i know that n should be large I'm not entirely certain it should be quite so large.

Any suggestions?

 

[Jhero]

Thank you for your post. I am a scientist and I would like to offer my thoughts on this problem. Firstly, it is important to note that for a system to exhibit quantum behavior, it must have a small enough wavelength compared to its size. In this case, the pendulum of a grandfather clock is relatively large compared to the size of an atom, so we can expect the quantum amplitudes to be negligible.

However, if we were to consider the pendulum as a quantum system, we could estimate the value of n by looking at the energy levels of the system. As you mentioned, the energy levels are given by En = (n + 1/2) hbarw. In this case, w is the angular frequency of the pendulum, which can be calculated using the period T = 1 s. This gives us w = 2pi/T = 2pi rad/s.

Now, to calculate the potential energy at the highest point of the oscillation, we can use the equation PE = mgh, where m is the mass of the bob, g is the acceleration due to gravity, and h is the height of the pendulum. From the given information, we can calculate that h = 3 cm = 0.03 m. Plugging in the values, we get PE = (0.2 kg)(9.8 m/s^2)(0.03 m) = 0.0588 J.

Equating this potential energy to the energy level, we get 0.0588 J = (n + 1/2)(1.05x10^-34 J*s)(2pi rad/s). Solving for n, we get n = 0.0588/(1.05x10^-34 x 2pi) - 1/2 = 2.8x10^33. This value may seem large, but it is not unexpected for a macroscopic object like a pendulum.

In conclusion, while it is unlikely for the pendulum of a grandfather clock to exhibit significant quantum behavior, if we were to consider it as a quantum system, we can estimate the value of n to be around 2.8x10^33. I hope this helps. Please let me know if you have any further questions or concerns.