# Quick QM Question Homework: Find Probability & Average Energy

• iamalexalright
In summary, the conversation discusses finding the state of a particle at a later time, the probability of measuring a particular energy value, and the average energy of the particle in terms of its energy eigenvalues. The key concept is that the wave function \Psi(x,t) can be written as a linear combination of the energy eigenfunctions \varphi_{n}(x) with complex coefficients b_n. The probability of measuring a specific energy value is equal to the square of the coefficient b_n.
iamalexalright

## Homework Statement

Given an initial ave function $$\Psi(x)$$ at t=0, and a complete set of energy eigenfunctions $$\varphi_{n}(x)$$ with corresponding eigenenergies E_n for a particle, and no other information, in therms of the given find:

a. Find the state of the particle at a later time

Solution: $$\Psi(x,t) = T(t)*\varphi_{n}(x) = e^{-iE_{n}t/\hbar}\varphi_{n}(x)$$

b. what is the probability that a measurement of energy will yield a particular value E_n?
$$<E> = \sum_{n}C_{n}P(C_n)$$
Where P is the probability but I don't know where to go from here.

c. Find an expression for the average energy of the particle in terms of the enrgy eigenvalues:
$$<E>=1/n \sum_{i=1}^{n}E_{n}$$

All seems right besides b? Any hints on b?

One important point you're missing is that $$\psi$$ can be any linear combination of the energy eigenfunctions $$\phi_n$$. Try the problem armed with the knowledge that:

$$\Psi(x,t) = \Sigma_n \phi_n(x,t)$$

I would be glad to answer any further questions, but this seemed to be the major stumbling block.

Last edited:
oh, duh...

a.
$$\Psi(x) = \sum b_{n}\varphi_{n}(x)$$

which implies

$$\Psi(x,t) = \sum b_{n}e^{-iE_{n} t/ \hbar} \varphi_{n}$$

Which means in part b the probability of yielding E_n is simply b_n, correct?

$$b_{n} = <\varphi_{n}|\Psi(x)>$$

hrm, LaTeX is either messing up for me or I missed something in my formatting but hopefully you can see what I meant for Psi in my second post

Note that
$$b_{n} = <\varphi_{n}|\Psi(x)>$$
Can be a complex number. Does a complex probably mean anything? Hint hint...

## 1. What is the purpose of "Quick QM Question Homework: Find Probability & Average Energy"?

The purpose of this homework is to test your understanding of basic concepts in quantum mechanics, specifically finding probabilities and average energy values.

## 2. What topics are covered in this homework?

This homework covers topics such as probability distributions, wave functions, expectation values, and the Schrodinger equation.

## 3. How do I find the probability of a particle being in a certain state?

To find the probability of a particle being in a certain state, you need to square the magnitude of the wave function for that state and then normalize it by dividing by the total probability of all possible states.

## 4. How do I calculate the average energy of a particle?

The average energy of a particle can be calculated by taking the expectation value of the Hamiltonian operator, which is given by the integral of the wave function multiplied by the Hamiltonian operator.

## 5. Are there any resources available to help with this homework?

Yes, there are many resources available online that can help with understanding and solving problems related to quantum mechanics. You can also reach out to your instructor or classmates for help and clarification on any concepts or problems.

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