What is the expectation value of ψ = x3 when 0≤ x ≤a and 0 otherwise?

In summary, the conversation is about finding the expectation value <x> for a given wavefunction ψ = x3 when 0≤ x ≤a and 0 otherwise. The equation used to find this value is ∫ψ* x ψ dx = 1, where ψ* represents the complex conjugate of ψ. By normalizing the wavefunction and solving for A, the value of <x> is found to be (7/8)a, which is closer to a than 0.
  • #1
leroyjenkens
616
49

Homework Statement


ψ = x3 when 0≤ x ≤a and 0 otherwise
find <x>

Homework Equations


∫ψ*ψdx=1

The Attempt at a Solution


So first I multiplied x3 times A, to get Ax3, then plugging that into the equation, I get ∫A2x6dx=1
Then I solve that for A, getting A = [tex]\sqrt{\frac{7}{a^{7}}}[/tex]
So I plug that value back into my integral in place of A and since I'm solving for the expectation value <x>, I plug x into that equation too? This is confusing.
 
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  • #2
You have succefully normalised the wavefunction: with the A value that you calculated, the (total) probability of finding the particle anywhere is 1.
You still need now to calculate <x> = ∫ψ* x ψ dx .

You should find something between 0 and a, and closed to a than to 0.
 
  • #3
I'm sort of confused on what to plug in for ψ and ψ* in that equation. Ax3 for both? In that case, I get (7/8)a. That seems a little high. Thanks.
 

FAQ: What is the expectation value of ψ = x3 when 0≤ x ≤a and 0 otherwise?

What is the concept of expectation value in science?

Expectation value is a statistical measure used in science to determine the average value of a certain variable or quantity based on a probability distribution. It represents the theoretical or predicted value of a measurement or outcome.

How is expectation value calculated?

Expectation value is calculated by multiplying the possible values of a variable by their corresponding probabilities and then summing them together. This can be represented mathematically as E(x) = ∑(x * P(x)), where x is the variable and P(x) is the probability of that variable occurring.

What is the significance of expectation value in scientific research?

Expectation value is an important concept in scientific research as it allows scientists to make predictions and draw conclusions based on probability. It also helps in determining the most likely outcome of an experiment or observation.

Can expectation value be used to make accurate predictions?

While expectation value provides a theoretical value based on probability, it does not guarantee the exact outcome of a measurement or experiment. It is a statistical measure and therefore, the actual results may vary.

How does the concept of expectation value apply to quantum mechanics?

In quantum mechanics, expectation value is used to calculate the average value of physical quantities such as energy, momentum, and position. It plays a crucial role in understanding the behavior of particles at a quantum level and making predictions about their properties.

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