Students compute inner products to expand a wave function in a sinusoidal basis set. This activity introduces the inner product for wave functions, and the idea of approximating a wave function using a finite set of basis functions.
A basis set is a choice of orthonormal vectors to use as a representation. The basis set you're probably most familiar with is the cartesian basis in three dimensions, consisting of \(\hat x\), \(\hat y\), and \(\hat z\). The basis allows us, for instance, to represent states as a column vector: \begin{align} \vec v &= v_x \hat x + v_y \hat y + v_z \hat z \\ &\,\dot= \begin{pmatrix}v_x\\v_y\\v_z\end{pmatrix} &\text{where }v_x &= \hat x\cdot \vec v\text{ etc.} \end{align} We can express same concept in bra-ket notation a bit more generally \begin{align} |v\rangle &= \sum_n v_n|n\rangle &\text{where }v_n &= \langle n | v\rangle\text{ etc.} \end{align} where the sum runs over all possible basis vectors.
In the same way that we express a vector as a linear combination of basis vectors, we can use a linear combination of basis functions to describe the state of a particle in space. When we use Dirac notation, the math in fact looks identical to what you will be doing with spins: \begin{align} |\psi\rangle &= \sum_n C_n |n\rangle & C_n &= \langle n|\psi\rangle \end{align} where \(\left|{n}\right\rangle \) represents a basis state. To compute these expansion coefficients, you will need to know how to compute the inner product between wavefunctions.
The inner product between wave functions in one dimension is given by \begin{align} \langle \phi|\psi\rangle &= \int \phi(x)^*\psi(x) dx \end{align} where \(\phi(x)\) is the wavefunction representation of \(\left|{\phi}\right\rangle \), and \(\psi(x)\) is the wavefunction representation of \(\left|{\psi}\right\rangle \). The integral goes over the domain of the problem (in our case from \(0\rightarrow L\)). This inner product is similar to the inner product you are accustomed to in spin systems, with the summation turned into an integration due to the fact that there are an infinite number of positions possible.
The basis set we will use is a sinusoidal basis set, defined by \begin{align} |n\rangle \,\,\dot=\,\, \phi_n(x) = \sqrt{\frac{2}{L}}\sin(n\pi x/L) \end{align} where \(n\) is a positive integer. Note that when we use “integer-sounding” variables in a ket (like \(\left|{n}\right\rangle \) or \(\left|{m}\right\rangle \)), we mean basis states, and their wavefunction representation is denoted by \(\phi_n(x)\) or \(\phi_m(x)\).
Over time you'll get to discover some fun and interesting properties of this basis, and you'll also be using it a lot for classical physics in Oscillations and Waves next quarter.
One way that we frequently represent states in quantum mechanics is as a sum of basis states. This hearkens back to classical mechanics (and Static Fields) where you learned to write a vector in terms of basis vectors \begin{align} \vec r &= x\hat x + y \hat y + z\hat z \end{align} where you may recall that \(x = \vec r\cdot\hat x\), etc. In bra-ket language, we can write this relationship as \begin{align} |\psi\rangle &= \langle 1|\psi\rangle\, |1\rangle + \langle 2|\psi\rangle\, |2\rangle + \langle 3|\psi\rangle\, |3\rangle + \cdots \\ &= \sum_{n=1}^{\infty} \langle n|\psi\rangle\, |n\rangle \\ &= \sum_{n=1}^{\infty} C_n |n\rangle \end{align} Now this infinite sum doesn't look so convenient. Fortunately, we can approximate the sum by summing over a finite number of basis functions, and check that we are approaching the limit by seeing how our result changes when we increase the number of terms retained in the sum.
You will be given one of the following wave functions: \begin{align} \psi_1(x) &= \frac{\sqrt{30}}{L^2\sqrt{L}}x(x-L) & \psi_2(x) &= \frac{x^6\sin(\pi x/L)}{0.07931977085 L^6\sqrt{L}} \\ \psi_3(x) &= \frac{\sqrt{105}}{L^3\sqrt{L}}x^2(x-L) & \psi_4(x) &= \frac{\left(e^{x/L}-1\right)\left(e^{(x-L)/L}-1\right)}{0.1937570896322709\sqrt{L}} \\ \psi_5(x) &= \frac{\sqrt{495}}{L^5\sqrt{L}}x(x-L)^4 \end{align}