- #1

tanaygupta2000

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- Homework Statement
- Let ψ₀(x) and ψ₁(x) be the wave functions corresponding to the ground state and the first excited states of a one dimensional harmonic oscillator respectively. Consider the normalized state ϕ(x) = αψ₀(x) + βψ₁(x), where α and β are real numbers. The values of α and β for which <x>, the average value of the position is a minimum are:

(a) α = -β = 1/√2

(b) α = β = 1/√2

(c) α = 1/√3 and β = -√(2/3)

(d) α = 1/√3 and β = √(2/3)

- Relevant Equations
- For one-dimensoinal harmonic oscillator,

Ground state wavefunction, ψ₀(x) = (mw/πℏ)^(1/4) exp[-(mw/2ℏ)x^2]

First excited state wavefunction, ψ₁(x) = (mw/πℏ)^(1/4) * x√(2mw/ℏ) * exp[-(mw/2ℏ)x^2]

Average value of position, <x> = ∫xϕ(x) dx

After getting the values of ψ₀(x) and ψ₁(x), I put them in the expression of ϕ(x) to get:

ϕ(x) = (mw/πℏ)^(1/4) * exp[-(mw/2ℏ)x^2] * [α + βx√(2mw/ℏ)]

Now when attempting to find the value of <x> by ∫xϕ(x) dx, I am having trouble determining the limits, as I am getting nothing useful by integrating from -∞ to ∞.

Also, do I even need to find <x>, can't I equate the derivative of simply ϕ(x) to 0 to get the minima, as <x> ∝ ϕ(x) which is relatively easier to differentiate?

Kindly help !

ϕ(x) = (mw/πℏ)^(1/4) * exp[-(mw/2ℏ)x^2] * [α + βx√(2mw/ℏ)]

Now when attempting to find the value of <x> by ∫xϕ(x) dx, I am having trouble determining the limits, as I am getting nothing useful by integrating from -∞ to ∞.

Also, do I even need to find <x>, can't I equate the derivative of simply ϕ(x) to 0 to get the minima, as <x> ∝ ϕ(x) which is relatively easier to differentiate?

Kindly help !