HW 7: The Fundamental Postulates of QM

Question-1¶

Determine which of the followng functions is acceptable as a quantum wave function?

• $\frac{1}{x}$ on a range $(0,+\infty)$

• $e^{-x}cos(x)$ on a range $(0,+\infty)$

• $x^{10000}e^{-x}$ on a range $(0,+\infty)$

• $e^x$ on a range $(-\infty,+\infty)$

Question-2¶

Normalize the following wave functions:

• $\psi(x)=e^{-3x}$

• $\psi(x)=2e^{-3x}+3e^{-x}$

Question-3¶

1. Evaluate the commutator $[\hat{A},\hat{B}]$ for the following pair of $\hat{A}$ and $\hat{B}$ operators.

• $\hat{A}=\frac{d^2}{dx^2}$ and $\hat{B}=x$

• $\hat{A}=\frac{d}{dx}+x$ and $\hat{B}=\frac{d}{dx}-x$

• $\hat{A}=\frac{d}{dx}$ and $\hat{B}=\hat{A}^2$

2. What can we say about the measruments of correpsonding quantities A and B?

3. Can you write down eigenfunctions for A and B?

Question-4¶

1. Which of the following operators is Hermitian

• $\hat{A}=\frac{d^2}{dx^2}$

• $\hat{A}=x\frac{d}{dx}$

• $\hat{A}=i\frac{d^2}{dx^2}$

• $\hat{A}=x^2$

2. Show that sum of two hermitian operators $\hat{A}$ and $\hat{B}$ is also hermitian.

3. In what situation will the product of two hermitian operators $\hat{A}\hat{B}$ be hermitian itself?

Question-5¶

You are given the following two wave-functions that describes the state of quantum particle in a 1D box of size L. What would you get when you start measuring the value of energy in an experiment. Compute the spread of E values in your measurments $\sigma_E^2$.

• $\psi(x)=10sin\Big(\frac{2\pi x}{L} \Big)$

• $\psi(x)=sin\Big(\frac{1\pi x}{L} \Big)+ 2sin\Big(\frac{2\pi x}{L} \Big)$

Hint: These wave-functtions are not normalized. You must first normalize wave-functions before you can do any calculations with them.