Probability density for observable with continuous Spectrum

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Naarogaut
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Misplaced Homework Thread
I'm given a wave function for an electron which is given as:
1684834653817.png

For an electron in this state the kinetic energy is being measured, where the kinetic energy operator is p^2/2m. How can I find the probability (density) that an electron is found to have kinetic energy in the interval [E, E+dE]? I was thinking about using the born rule, but I am struggeling to use it for the infinite dimensional Hilbert space, since the eigenstates of the kinetic energy operator degenerate as far as I can tell...
 
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How about calculating ##\psi(p)##?
 
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FAQ: Probability density for observable with continuous Spectrum

What is a probability density function (PDF) in the context of a continuous spectrum?

A probability density function (PDF) in the context of a continuous spectrum describes the likelihood of a continuous random variable taking on a specific value. Unlike a discrete probability distribution, the PDF provides a density rather than a direct probability, and the probability of the variable falling within a particular range is obtained by integrating the PDF over that range.

How do you interpret the area under the PDF curve?

The area under the PDF curve between two points represents the probability that the random variable falls within that range. The total area under the entire PDF curve is equal to 1, reflecting the fact that the random variable must take on some value within the spectrum.

What is the difference between a PDF and a cumulative distribution function (CDF)?

The PDF provides the density of the probability at each point in the continuous spectrum, while the cumulative distribution function (CDF) gives the probability that the random variable is less than or equal to a certain value. The CDF is obtained by integrating the PDF from the lower bound of the spectrum up to the point of interest.

How is the expectation value calculated for an observable with a continuous spectrum?

The expectation value (or mean) of an observable with a continuous spectrum is calculated by integrating the product of the variable and its PDF over the entire range of the variable. Mathematically, it is expressed as \( E[X] = \int_{-\infty}^{\infty} x f(x) \, dx \), where \( f(x) \) is the PDF of the variable \( X \).

What role does normalization play in probability density functions?

Normalization ensures that the total area under the PDF curve is equal to 1, which corresponds to the total probability of all possible outcomes being 1. This is a crucial property of any probability distribution, and for a PDF, it is mathematically represented as \( \int_{-\infty}^{\infty} f(x) \, dx = 1 \).

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