Basic QM - Trying to catch up in a class

[FrogPad]

I haven't taken one of the recommended courses for a class I am in, so I'm playing a little catch-up here.

Q: The wave function of an electron in a one-dimensional infinite square well of width a,x -> (0,a) at time t=0 is given by:

$$\psi (x,0) \sqrt{\frac{2}{7}}\psi_1(x) + \sqrt{\frac{5}{7}}\psi_2(x)$$

where $$\psi_1(x)$$ and $$\psi_2(x)$$ are the ground and first excited stationary state of the system respectively, $$\psi_n(x)=\sqrt{\frac{2}{a}}\sin (n\pix/a), \,\,\, E_n = n^2\pi^2 \bar h^2/(2ma^2), \,], n=1,2,...$$

a) Write down the wavefunction at time t in terms of $$\psi_1(x)$$ and $$\psi_2(x)$$.

b) You measure the energy of an electron at time t=0. Write down possible values of the energy and the probability of measuring each.

c) Calculate the expectation value of hte enrgy in the state $$\psi(x,t)$$ above.

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I am lost here. I am stuck on (a) right now, and really do not know where to go. To me it looks like $$\psi_n(x)$$ is the wavefunction already solved from the schrodinger equation (with the boundary conditions 0 and a). So I need to express the total wavefunction in terms of two states psi_1, and psi_2. This to me seems like another condition that would come from a differential equation something with a time dependence. I am lost...

Would someone nudge me in a proper direction?

thanks,

 

[dextercioby]

Hint for 1. $$\psi (t)=U(t)\psi=e^{\frac{1}{i\hbar}tH} \psi$$

Daniel.

 

[m2003]

I understand that catching up in a class can be challenging, especially when it involves complex concepts like quantum mechanics. Let me try to provide some guidance on the problem you presented.

Firstly, you are correct in your understanding that the wave function given in the problem is a linear combination of the ground state and first excited state of the system. This means that the total wave function can be expressed as a sum of these two individual wave functions.

To solve part (a) of the problem, you simply need to write out the wave function at time t, which is given by \psi(x,t) = \sqrt{\frac{2}{7}}\psi_1(x) e^{-iE_1t/\hbar} + \sqrt{\frac{5}{7}}\psi_2(x) e^{-iE_2t/\hbar}. This is obtained by using the time-dependent Schrodinger equation, which incorporates the time dependence of the wave function.

For part (b), you are asked to write down the possible values of energy and their corresponding probabilities at time t=0. Using the wave function given in the problem, you can calculate the energy by taking the inner product of the wave function with the energy operator. This will give you two possible values of energy, corresponding to the two states in the wave function.

As for part (c), you are asked to calculate the expectation value of the energy in the state \psi(x,t) . This can be done by taking the inner product of the wave function with the energy operator, and then dividing by the norm of the wave function squared.

I hope this helps you in your understanding of the problem. Remember, don't hesitate to seek help from your instructor or classmates if you are still struggling. Good luck!