Quantum harmonic oscillator

• quietrain
In summary, you can use the angular momentum formula to find the energy of a quantum harmonic oscillator, but it is not always the same as twice the energy of a single linear oscillator.

Homework Statement

Consider a harmonic oscillator with mass=0.1kg, k=50N/m , h-bar=1.055x10-34
Let this oscillator have the same energy as a mass on a spring, with the same k and m, released from rest at a displacement of 5.00 cm from equilibrium. What is the quantum number n of the state of the harmonic oscillator?

The Attempt at a Solution

ok, so i was wondering, why can't i use angular momentum quantization?

meaning mvr = n(h-bar)
so i divide both sides by r2
i get m(k/m)-1/2=n(h-bar)/r2

make n subject, sub in all values, i get 5.29*1031.. which happens to be twice the correct answer.

so did i make a careless mistake somewhere? or was i never been able to use this formula? thanks!

Before applying a formula, think about the meaning of the symbols. mvr is the angular momentum of a point mass orbiting along a circle of radius r. But here the body oscillates along a straight line.

The energy of a quantum harmonic oscillator is

$$E=\hbar \omega (v+1/2)$$

v is the vibrational quantum number, a non-negative integer.

ehild

hmm... so you mean that using my attempt's formula is totally wrong in this case?

but is there a reason why i get exactly twice the correct answer?

is it coincidence? or is there some underlying principle?

A circular motion can be considered as two perpendicular harmonic oscillators.

oh. but why?

what is the quantity that relates them both?

is it the distance? the speed? the momentum? or?

Let a point mass vibrate along the x direction according to the formula x=A cos (wt), and at the same time it moves and along the y direction: y=A sin(wt). The resultant trajectory is a circle of radius A,
and the body moves with angular velocity w and linear velocity v=Aw. The angular momentum is

$$A^2\omega m=\hbar n$$.

The energy of the circular motion is the sum of the kinetic energy and the potential energy, and it is the same as twice the energy of a single linear oscillator.

$$0.5v^2 m+0.5 kA^2 = m A^2 \omega ^2=\hbar \omega (n+0.5)$$

n is so big that the 1/2 term is negligible, so you have the same quantum numbers from the energy of two harmonic oscillators as from the angular momentum.

ehild

ehild said:
Let a point mass vibrate along the x direction according to the formula x=A cos (wt), and at the same time it moves and along the y direction: y=A sin(wt). The resultant trajectory is a circle of radius A,
and the body moves with angular velocity w and linear velocity v=Aw. The angular momentum is

$$A^2\omega m=\hbar n$$.

The energy of the circular motion is the sum of the kinetic energy and the potential energy, and it is the same as twice the energy of a single linear oscillator.

$$0.5v^2 m+0.5 kA^2 = m A^2 \omega ^2=\hbar \omega (n+0.5)$$

n is so big that the 1/2 term is negligible, so you have the same quantum numbers from the energy of two harmonic oscillators as from the angular momentum.

ehild

oh, so is it right to say that KE of 1 linear oscillator = 1/2 mv^2 ?so we assume KE of single linear oscillator = PE of it. so we have 2 times?

so total energy of circular motion = total energy of oscillator?

quietrain said:
oh, so is it right to say that KE of 1 linear oscillator = 1/2 mv^2 ?so we assume KE of single linear oscillator = PE of it. so we have 2 times?

so total energy of circular motion = total energy of oscillator?

This is not quite true. The total energy is KE +PE, but when one of them is maximum, the other is zero.
For a body preforming SHM, x=Asin (wt). The potential energy is maximum at x=A, and PE(max) = 1/2kA^2. The speed is maximum at x=0, and v(max)=Aw, the maximum kinetic energy = 1/2 m A^2 w^2. As w^2=k/m, KE(max)=1/2 kA^2, the same as the maximum potential energy.

Two orthogonal vibrations with the same frequency and amplitude and phase difference of pi/2 between them is equivalent with a circular motion. Because of the phase difference, when one oscillator has maximum potential energy, the other has its maximum kinetic energy, but they are equal, so the energy of this circular motion is equal to that of two oscillators.

Note that this is true only for such circular motions which result from these two vibrations. For the planetary motion, or motion in Coulomb field, both the force and the energy is different.

ehild

oh i see! so when i use mvr = nh/2pi , i am acutally accounting for both the KEmax and the PEmax of the circular motion? that's why i have to divide by 2 if i want to find just one oscillator?

Something like that :). I think you haven't studied quantum theory yet, have you? Because the derivation of the energy of quantum harmonic oscillator is at advanced level. Perhaps the formula was shown already during the classes.

ehild

yup thanks a lot!

What is a quantum harmonic oscillator?

A quantum harmonic oscillator is a simplified model used to describe certain physical systems, such as atoms and molecules, that vibrate or oscillate at specific frequencies. It is a quantum mechanical system, meaning it takes into account both the wave-like and particle-like nature of matter.

What are the key features of a quantum harmonic oscillator?

The key features of a quantum harmonic oscillator include a potential energy that increases quadratically with distance from the center, a discrete set of energy levels, and a characteristic frequency of oscillation. It also exhibits zero-point energy, meaning it cannot have a ground state with zero energy.

What is the equation for the energy levels of a quantum harmonic oscillator?

The energy levels of a quantum harmonic oscillator are given by the equation En = (n + 1/2)ħω, where n is the quantum number that represents the energy level, ħ is the reduced Planck's constant, and ω is the angular frequency of oscillation.

How does the quantum harmonic oscillator relate to the Heisenberg uncertainty principle?

The Heisenberg uncertainty principle states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. In the case of a quantum harmonic oscillator, this principle is reflected in the fact that the uncertainty in the position and momentum of the oscillator's ground state is not equal to zero, unlike in classical mechanics.

What are some real-world applications of the quantum harmonic oscillator?

The quantum harmonic oscillator has many applications in physics, chemistry, and engineering. For example, it is used to model the behavior of atoms and molecules, design electronic circuits, and understand the properties of materials such as crystals and polymers. It also plays a crucial role in the development of quantum technologies, such as quantum computing and quantum cryptography.