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kodel420
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- Homework Statement
- I am currently studying an Introductory Quantum Mechanics course, and one past example question talks about the following. An infinite potential well given by $$V(x) = \begin{cases} 0 \; -\frac{L}{2} \le x \le \frac{L}{2} \\ \infty \; \text{otherwise} \end{cases}$$
At time ## t=0 ##, the particle is in the ground state of the well and the well is doubled in width and now runs from ## -L ## to ## L ##. What is the probability that the particle is found in the new n=2 state [answer : ## \frac{1}{\sqrt{2}} ##].
- Relevant Equations
- $$V_i(x) = \begin{cases} 0 \; -\frac{L}{2} \le x \le \frac{L}{2} \\ \infty \; \text{otherwise} \end{cases}$$
$$V_f(x) = \begin{cases} 0 \; -L \le x \le L \\ \infty \; \text{otherwise} \end{cases}$$
Firstly I have found the eigenstates for both the original well and the new well as the following
$$\psi_{n,\frac{L}{2}} = \begin{cases} \sqrt{\frac{2}{L}} \cos{\frac{n \pi x}{L}} \; \; \; \; \; n \text{ odd} \\ \sqrt{\frac{2}{L}} \sin{\frac{n \pi x}{L}} \; \; \; \; \; n \text{ even} \end{cases}$$
$$\psi_{n,L} = \begin{cases} \frac{1}{\sqrt{L}} \cos{\frac{n \pi x}{2L}} \; \; \; \; \; n \text{ odd} \\ \frac{1}{\sqrt{L}} \sin{\frac{n \pi x}{2L}} \; \; \; \; \; n \text{ even} \end{cases}$$
Then using the principle of superposition the original ground state can be written as a superposition of the new eigenstates
$$\psi_{1,\frac{L}{2}} = \sum_{n = 1}^{\infty} {c_n}\psi_{n,L} \; \; \; \; \; c_n = \int_{-L/2}^{L/2} \psi_{1,\frac{L}{2}} \; \psi_{n,L} \;dx$$
and by the Born Rule the probability is just ## |c_n|^2 ## so we are looking to find ## c_2 ##. My problem is that the integral $$\int_{-L/2}^{L/2} \psi_{1,\frac{L}{2}} \; \psi_{2,L} = 0$$ obviously not giving the desired result. So my question is, am I doing something wrong, or does the symmetry of the well cause an issue, because the same issue does not arise for a well from ## 0 ## to ## L ## being extended to ## 0 ## to ## 2L ##
$$\psi_{n,\frac{L}{2}} = \begin{cases} \sqrt{\frac{2}{L}} \cos{\frac{n \pi x}{L}} \; \; \; \; \; n \text{ odd} \\ \sqrt{\frac{2}{L}} \sin{\frac{n \pi x}{L}} \; \; \; \; \; n \text{ even} \end{cases}$$
$$\psi_{n,L} = \begin{cases} \frac{1}{\sqrt{L}} \cos{\frac{n \pi x}{2L}} \; \; \; \; \; n \text{ odd} \\ \frac{1}{\sqrt{L}} \sin{\frac{n \pi x}{2L}} \; \; \; \; \; n \text{ even} \end{cases}$$
Then using the principle of superposition the original ground state can be written as a superposition of the new eigenstates
$$\psi_{1,\frac{L}{2}} = \sum_{n = 1}^{\infty} {c_n}\psi_{n,L} \; \; \; \; \; c_n = \int_{-L/2}^{L/2} \psi_{1,\frac{L}{2}} \; \psi_{n,L} \;dx$$
and by the Born Rule the probability is just ## |c_n|^2 ## so we are looking to find ## c_2 ##. My problem is that the integral $$\int_{-L/2}^{L/2} \psi_{1,\frac{L}{2}} \; \psi_{2,L} = 0$$ obviously not giving the desired result. So my question is, am I doing something wrong, or does the symmetry of the well cause an issue, because the same issue does not arise for a well from ## 0 ## to ## L ## being extended to ## 0 ## to ## 2L ##
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