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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1. What is probability density for observable with continuous spectrum?

The probability density for an observable with continuous spectrum is a mathematical function that describes the likelihood of obtaining a particular measurement or value for that observable. It is represented by a probability density function (PDF) and is used in quantum mechanics to predict the outcomes of measurements on a quantum system.

2. How is probability density for observable with continuous spectrum different from discrete spectrum?

The main difference between probability density for observable with continuous spectrum and discrete spectrum is the type of values that the observable can take. In discrete spectrum, the observable can only take on a finite or countably infinite number of values, while in continuous spectrum, the observable can take on any value within a certain range. This means that the probability density for an observable with continuous spectrum is a continuous function, while for a discrete spectrum, it is a discrete function.

3. What is the role of the wavefunction in probability density for observable with continuous spectrum?

The wavefunction is a mathematical function that describes the quantum state of a system. In probability density for observable with continuous spectrum, the wavefunction is used to calculate the probability density function, which in turn is used to determine the likelihood of obtaining a particular measurement for the observable. The wavefunction also plays a crucial role in determining the energy levels and transitions of a quantum system.

4. How is probability density for observable with continuous spectrum used in quantum mechanics?

Probability density for observable with continuous spectrum is an essential concept in quantum mechanics as it allows us to make predictions about the outcomes of measurements on a quantum system. By calculating the probability density function using the wavefunction, we can determine the likelihood of obtaining a particular value for the observable. This information is then used to make predictions about the behavior and properties of quantum systems.

5. Can probability density for observable with continuous spectrum change over time?

Yes, the probability density for an observable with continuous spectrum can change over time. This is because the wavefunction, which is used to calculate the probability density function, can evolve and change as the quantum system evolves. This change in the wavefunction can lead to changes in the probability density for the observable, which in turn can affect the outcomes of measurements on the system.

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