Hilbert space and the probability to find the particle right of the origin

We study a quantum particle confined to a one-dimensional box with walls at the positions $$\pm 1$$. The Hilbert-space of this system in the Schrödinger representation is hence given by $$L_2([-1; 1])$$ with the scalar product $$\left \langle f,g \right \rangle=\int_{-1}^{1}\overline{f(x)}g(x) \,\mathrm{d}x,\quad f,g\in L_2([-1,1])$$ In this Hilbert space, we consider the two functions $$f_0(x) := (1+i) \exp(i\pi x)$$ and $$f_1(x):=\exp(i2\pi x)$$.

Now the problems are as follows

1) We now consider a general superposition $$\Psi=\alpha\phi_0+\beta\phi_1$$ of the two states $$\phi_0:=f_0/2$$ and $$\phi_1:=f_1/\sqrt{2}$$. Show that in order for $$\Psi$$ to be normalized we need $$|\alpha|^2+|\beta|^2=1$$.

2) Determine the probability to find the particle right of the origin if it is in the state $$\Psi$$, depending on the two parameters $$\alpha=\left \langle \phi_0,\Psi \right \rangle$$ and $$\beta=\left \langle \phi_1,\Psi \right \rangle$$

3) Find values for $$\alpha$$ and $$\beta$$ that maximize the probability to find the particle right of the origin. Hint: By using that $$\Psi$$ and $$e^{i\alpha}\Psi$$ represent the same physical state of the system, we can choose $$0\leq \alpha\leq 1$$ and $$\beta=\sqrt{1-\alpha^2}e^{i\eta}$$.

I have no problem with number one. I have tried with the second one, but I am not sure if I have reached the right conclusion. Because of that, I can not continue with the last problem.

2) We have to determine $$\int_{0}^{1}|\Psi(x)|^2\,\mathrm{d}x$$. First, we observe that $$|\Psi|^2=|\alpha|^2|\phi_0|^2+\alpha \bar{\beta}\overline{\phi_1}\phi_0+ \bar{\alpha}\beta\overline{\phi_0}\phi_1+|\beta|^2|\phi_1|^2$$ then $$\int_{0}^{1} |\Psi(x)|^2\,\mathrm{d}x=|\alpha|^2\int_{0}^{1}|\phi_0(x)|^2\,\mathrm{d}x+\alpha \bar{\beta}\int_{0}^{1}\overline{\phi_1(x)}\phi_0(x)\,\mathrm{d}x\\+ \bar{\alpha}\beta\int_{0}^{1}\overline{\phi_0(x)}\phi_1(x)\,\mathrm{d}x+|\beta|^2\int_{0}^{1}|\phi_1(x)|^2\,\mathrm{d}x$$ The first and the last integrals are $$1/2$$. The two other are $$\int_{0}^{1}\overline{\phi_1(x)}\phi_0(x)\,\mathrm{d}x=\frac{1+i}{2\sqrt{2}}\int_{0}^{1} e^{-i\pi x}\,\mathrm{d}x=\frac{1-i}{\sqrt{2}\pi},$$ $$\int_{0}^{1}\overline{\phi_0(x)}\phi_1(x)\,\mathrm{d}x=\frac{1-i}{2\sqrt{2}}\int_{0}^{1} e^{i\pi x}\,\mathrm{d}x=\frac{i+1}{\sqrt{2}\pi}$$ Altogether, we get $$\int_{0}^{1} |\Psi(x)|^2\,\mathrm{d}x=\frac{1}{2}\left ( |\alpha|^2+|\beta|^2 \right )+\alpha \bar{\beta}\left ( \frac{1-i}{\sqrt{2}\pi} \right )+ \bar{\alpha}\beta\left ( \frac{i+1}{\sqrt{2}\pi} \right ).$$ Since $$\alpha \bar{\beta}(1-i)+ \bar{\alpha}\beta(1+i) =\alpha \bar{\beta}(1-i)+ \overline{\alpha\bar{\beta}(1-i)} =2\Re(\alpha \bar{\beta}(1-i))$$ and with 1), we then get $$\int_{0}^{1} |\Psi(x)|^2\,\mathrm{d}x=\frac{1}{2}+\frac{2}{\sqrt{2}\pi}\Re(\alpha \bar{\beta}(1-i)).$$ Is this correct, or am I missing something?

• It looks right to me! – Adrian Keister Jan 30 at 20:51
• @AdrianKeister I'm glad to hear it... If you have time, could you please show me a way for the last problem? The hint doesn't seem helpful to me. – UnknownW Jan 30 at 21:24
• Hmm. Well, the maximization boils down to maximizing $\operatorname{Re}(\alpha\overline{\beta}(1-i)),$ subject to $|\alpha|^2+|\beta|^2=1.$ – Adrian Keister Jan 30 at 21:35
• @AdrianKeister That's true. I could write $|\beta|=\sqrt{1-|\alpha|^2}$, which implies $\beta=\sqrt{1-|\alpha|^2}e^{i\eta}$ for some number $\eta$. Should I then calculate the real part of $\alpha \sqrt{1-|\alpha|^2}e^{-i\eta}(1-i)$ and then finding the maximum of it with respect to $\alpha$, as long as I assume $\alpha$ is a real number? – UnknownW Jan 30 at 21:53

For problem 3), we need to maximize $$\operatorname{Re}(\alpha\overline{\beta}(1-i))$$ subject to $$|\alpha|^2+|\beta|^2=1$$. Let \begin{align*} \alpha&=r_{\alpha}e^{i\theta_{\alpha}} \\ \beta&=r_{\beta}e^{i\theta_{\beta}}, \\ \end{align*} which implies $$|\alpha|^2+|\beta|^2=r_{\alpha}^2+r_{\beta}^2=1,$$ and we're trying to maximize \begin{align*} \operatorname{Re}(\alpha\overline{\beta}(1-i))&=\operatorname{Re}(r_{\alpha}e^{i\theta_{\alpha}}r_{\beta}e^{-i\theta_{\beta}}\sqrt{2}\,e^{-i\pi/4}) \\ &=\sqrt{2}\,r_{\alpha}r_{\beta}\operatorname{Re}\left[e^{i(\theta_{\alpha}-\theta_{\beta}-\pi/4)}\right] \\ &=\sqrt{2}\,r_{\alpha}r_{\beta}\cos(\theta_{\alpha}-\theta_{\beta}-\pi/4). \end{align*} As the $$r$$'s show up symmetrically, choose $$r_{\alpha}=r_{\beta}=1/\sqrt{2},$$ and choose \begin{align*} \theta_{\alpha}-\theta_{\beta}-\frac{\pi}{4}&=0 \\ \theta_{\alpha}-\theta_{\beta}&=\frac{\pi}{4} \end{align*} to maximize the cosine.
One final point: once you do all this, you should definitely back-calculate the actual probability, and make sure it isn't bigger than $$1.$$ If it is, you can always dial back something (the angles would be the easiest) to make it smaller.
A quick calculation: $$\frac12+\frac{2}{\sqrt{2}\,\pi}(\sqrt{2}(1/2))=\frac12+\frac{1}{\pi}<1,$$ so you're safe.
• Wow, thank you. Quick question: Is it possible to choose, say $r_\alpha=1/2$, so that we have $r_\beta=1-1/4 = 3/4$? (I do not understand the symmetry part.) – UnknownW Jan 30 at 23:43