A 2.3-m-diameter water droplet is moving with a speed of 1.0 m/s in a 25-m-long

Question

A 2.3-m-diameter water droplet is moving with a speed of 1.0 m/s in a 25-m-long box.

a. Estimate the particle's quantum number. Express your answer using two significant figures.

b. Use the correspondence principle to determine whether quantum mechanics is needed to understand the particle's motion or if it is "safe" to use classical physics.

choice 1: In this case we can safely use classical physics to describe its motion.

choice 2: In this case quantum mechanics is needed to understand the particle's motion.

Explanation / Answer

a) A classical particle bouncing around in a box isn't described by a single energy eigenfunction--it's the superposition of a lot of states.

But I think they want you to just compare energy levels.

The energy of a particle-in-a-box is: (h-bar^2 pi^2 / 2mL^2) n^2

The kinetic energy of your particle is: 1/2 mv^2

Set those equal and solve for the quantum number:
n = sqrt[1/2 mv^2 (2mL^2 / h-bar^2 pi^2)]
n= (mvL / h-bar pi) =m(1 x 10-6 x 25 x 10-6 )/(6.6 x 10-34/2) = 7.58 x 1022 x m

By knowing m we can find n.

b) That's a very less-than-rigorous treatment, but I think it's what they intended by the problem just to show you that a classical particle has very very high quantum numbers, so it is effectively not quantized.

So,In this case we can safely use classical physics to describe its motion.

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