Wavefunction with specific values

[Gringema]

Homework Statement

Near a certain position x_A, a quanton's wavefunction ψ(x_A)=-0.3 in some units. Near a certain position x_B, a quanton's wavefunction ψ(xB[\SUB])=0.12 in the same units. If we do an experiment to locate the quanton, how many times more likely is i that the result is in a small range near x_A than in a range twice as large near x_B? Please explain.

Homework Equations

ψ(x)=ψ/sqrt(Δx)
|ψ|²=Pr(x)

The Attempt at a Solution

I used the definitions of wavefunction and amplitude to get Pr(x)=|ψ(x)|²Δx.
So Pr(x_A)=(-0.3)²=0.09
And Pr(x_B)=2(0.12)²=0.0288
Therefore Pr(x_A)=3.125Pr(x_B)

Is this the right way to think about this? And how would I "explain"?

 

[ehild]

Homework Statement

Near a certain position x_A, a quanton's wavefunction ψ(x_A)=-0.3 in some units. Near a certain position x_B, a quanton's wavefunction ψ(xB[\SUB])=0.12 in the same units. If we do an experiment to locate the quanton, how many times more likely is i that the result is in a small range near x_A than in a range twice as large near x_B? Please explain.

Homework Equations

ψ(x)=ψ/sqrt(Δx)
|ψ|²=Pr(x)

The Attempt at a Solution

I used the definitions of wavefunction and amplitude to get Pr(x)=|ψ(x)|²Δx.
So Pr(x_A)=(-0.3)²=0.09
And Pr(x_B)=2(0.12)²=0.0288
Therefore Pr(x_A)=3.125Pr(x_B)

Is this the right way to think about this? And how would I "explain"?

<sub>|ψ(x)|^2 is the probability density of finding the particle around some specified x. So the probability that a particle is in a narrow range of width Δx around x is P = |ψ(x)|^2 Δx.
So P(xA) = Δx(-0.3)²=0.09 Δx and Pr(xB)=2Δx(0.12)²=0.0288 Δx, but you got the ratio correctly.

ehild</sub>