# Quantum Harmonic Oscillator problem

• Caulfield
In summary, the probability of the particle being found within a distance of (mk)^(-1/4) *√ħ/2 of the origin is 1/2π^4 * e^-1.
Caulfield

## Homework Statement

For the n = 1 harmonic oscillator wave function, find the probability p that, in an experiment which measures position, the particle will be found within a distance d = (mk)-1/4√ħ/2 of the origin. (Hint: Assume that the value of the integral α = ∫0^1/2 x^2e^(-x2/2) dx is known and express your result as a function of α)

## Homework Equations

distance from 0 to d; d = (mk)^(-1/4) *√ħ/2

s=[km^(1/4) /ħ^2]*x

Normalize condition: Cn = 1/ (π^2√(2^nn!)

Harmonic Oscillator wave function for n = 1 ψ1 = C1(2s)e^-s2/2

## The Attempt at a Solution

Is the formula that I want to use here p=ψ∗(x)ψ(x) or
p=∫ψ∗(x)ψ(x)dx? (p stands for probability.)

s=[km^(1/4) /ħ^2]* (mk)^(-1/4) *√ħ/2 = 1/2

C1=1/π^2√2 so ψ1 = (1/π^2√2)e^-1/2

so, now, do I use:

p=ψ∗(x)ψ(x)= [(1/π^(2)√2)e^-1/2]^2= 1/2π^4 * e^-1

I believe that this is the right way, but the hint confuses me.

Caulfield said:

## Homework Statement

For the n = 1 harmonic oscillator wave function, find the probability p that, in an experiment which measures position, the particle will be found within a distance d = (mk)-1/4√ħ/2 of the origin. (Hint: Assume that the value of the integral α = ∫0^1/2 x^2e^(-x2/2) dx is known and express your result as a function of α)

## Homework Equations

distance from 0 to d; d = (mk)^(-1/4) *√ħ/2

s=[km^(1/4) /ħ^2]*x

Normalize condition: Cn = 1/ (π^2√(2^nn!)

Harmonic Oscillator wave function for n = 1 ψ1 = C1(2s)e^-s2/2

## The Attempt at a Solution

Is the formula that I want to use here p=ψ∗(x)ψ(x) or
p=∫ψ∗(x)ψ(x)dx? (p stands for probability.)

s=[km^(1/4) /ħ^2]* (mk)^(-1/4) *√ħ/2 = 1/2

C1=1/π^2√2 so ψ1 = (1/π^2√2)e^-1/2

so, now, do I use:

p=ψ∗(x)ψ(x)= [(1/π^(2)√2)e^-1/2]^2= 1/2π^4 * e^-1

I believe that this is the right way, but the hint confuses me.
I don't see how you got that expression for ##p##. What happened to the rest of the wave function?

I changed C1 when n=1 and s(d) when d = (mk)^(-1/4) *√ħ/2.

then ψ∗ψ=ψ^2 (since there is no i, ψ is real)

I made i typing mistake, I am sorry.

so ψ1 = C1(2s)e^-((s^2)/2)

Changing C1 and s gives ψ1 =(1/π^2√2)*(2*1/2)e^-((1/2^2)/2)

which gives ψ1 =(1/π^2√2)e^-1/8

and ψ1*ψ1=(1/2π^4)e^-1/4

Where's ##x##?

x is d.

Why are you plugging ##d## in for ##x##?

The problem asks you to find the probability the particle is found within a distance ##d## of the origin. What if it were to ask you find the probability the particle was found within a distance ##d/2## and ##d## of the origin? Would you still just plug in ##d## for ##x##? How would the ##d/2## fit in?

right... so now i see it makes more sense to use ∫ψ∗(x)ψ(x)dx... ok, i got the result. thanks

## 1. What is a Quantum Harmonic Oscillator?

The Quantum Harmonic Oscillator is a fundamental concept in quantum mechanics that describes the behavior of a particle in a potential well, similar to a mass on a spring. It is a model used to understand the energy levels and wave functions of a system in a quantum mechanical context.

## 2. What is the Schrödinger equation for the Quantum Harmonic Oscillator?

The Schrödinger equation for the Quantum Harmonic Oscillator is a second-order differential equation that describes the time evolution of the wave function of a system. It takes into account the potential energy of the system and results in a discrete set of energy levels and corresponding wave functions.

## 3. How does the Quantum Harmonic Oscillator differ from the classical Harmonic Oscillator?

The Quantum Harmonic Oscillator differs from the classical Harmonic Oscillator in that it takes into account the principles of quantum mechanics, such as the uncertainty principle and quantization of energy levels. In the classical Harmonic Oscillator, the energy levels are continuous and the particle can have any amount of energy, while in the quantum version, the energy levels are discrete and the lowest possible energy level is non-zero.

## 4. What is the significance of the zero-point energy in the Quantum Harmonic Oscillator?

The zero-point energy in the Quantum Harmonic Oscillator refers to the lowest possible energy level of the system, which is non-zero. This is a consequence of the quantization of energy levels in quantum mechanics and has important implications for the behavior of a particle in a potential well.

## 5. How is the Quantum Harmonic Oscillator used in real-world applications?

The Quantum Harmonic Oscillator has many practical applications in fields such as quantum computing, quantum chemistry, and material science. It is also used as a model for various physical systems, such as atoms, molecules, and crystals, to understand their energy levels and properties. Additionally, the principles of the Quantum Harmonic Oscillator are essential for understanding more complex quantum systems and phenomena.

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