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.
- The coefficients \(C_n\) are a representation of a quantum state.
- An expansion in a basis can be approximated by summing over only a few basis states, with the approximation getting better as we sum over more.
- For wave functions, the inner product is an integral.
- The coefficients in an expansion in a basis are inner products \(C_n = \langle n|\psi\rangle\).
- An expansion in a basis requires summing over all possible basis states (\(n=1\rightarrow \infty\) in our case).
- Functions can be orthogonal.
- Sometimes a wave function is orthogonal to a basis state, esp. when they have different symmetry.
- If we inexactly compute a number that is zero, we often get a result that is nonzero, but small.
- \(\langle n|m\rangle = \delta_{nm}\) for an orthonormal basis.
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}
I want students to learn to plot things "automatically" as part of checking whether their code is even working, but for now I still ask them to do so, since I expect that with this being quantum mechanics they might have more trouble activating their usual sense-making skills.
We have commonly seen students making \(x\) an input to their function to compute \(C_n\). They aren't yet clear on what an inner product is, and it can help to remind them of the definition of the inner product.
There is commented out here another version of this activity that we tried after students attempted the above.
keyboard Computational Activity
120 min.
quantum mechanics operator matrix element particle in a box eigenfunction
Students find matrix elements of the position operator \(\hat x\) in a sinusoidal basis. This allows them to express this operator as a matrix, which they can then numerically diagonalize and visualize the eigenfunctions.assignment Homework
Given the polar basis kets written as a superposition of Cartesian kets \begin{eqnarray*} \left|{\hat{s}}\right\rangle &=& \cos\phi \left|{\hat{x}}\right\rangle + \sin\phi \left|{\hat{y}}\right\rangle \\ \left|{\hat{\phi}}\right\rangle &=& -\sin\phi \left|{\hat{x}}\right\rangle + \cos\phi \left|{\hat{y}}\right\rangle \end{eqnarray*}
Find the following quantities: \[\left\langle {\hat{x}}\middle|{\hat{s}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{{\hat{s}}}\right\rangle ,\quad \left\langle {\hat{x}}\middle|{\hat{\phi}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{\hat{\phi}}\right\rangle \]
group Small Group Activity
30 min.
Taylor series power series approximation
This activity starts with a brief lecture introduction to power series and a short derivation of the formula for calculating the coefficients of a power series for a particular function:
\[c_n={1\over n!}\, f^{(n)}(z_0)\]
After a brief lecture deriving the formula for the coefficients of a power series, students compute the power series coefficients for a \(\sin\theta\) (around both the origin and \(\frac{\pi}{6}\)). The meaning of these coefficients and the convergence behavior for each approximation is discussed in the whole-class wrap-up.
accessibility_new Kinesthetic
10 min.
Quantum State Vectors Complex Numbers Spin 1/2 Arms Representation
Students, working in pairs, use their left arms to represent each component in a two-state quantum spin 1/2 system. Reinforces the idea that quantum states are complex valued vectors. Students make connections between Dirac, matrix, and Arms representation.group Small Group Activity
60 min.
Wavefunctions quantum states probability amplitude histograms matrix notation of quantum states Arms representation
Students review using the Arms representation to represent states for discrete quantum systems and connecting the Arms representation to histogram and matrix representation. The student then extend the Arms representation to begin exploring the continuous position basis.keyboard Computational Activity
120 min.
finite difference hamiltonian quantum mechanics particle in a box eigenfunctions
Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then usenumpy
to solve for eigenvalues and eigenstates, which they visualize.
assignment Homework
Writing an operator in matrix notation in its own basis is easy: it is diagonal with the eigenvalues on the diagonal.
What if I want to calculate the matrix elements using a different basis??
The eigenvalue equation tells me what happens when an operator acts on its own eigenstate. For example: \(\hat{S}_y\left|{\pm}\right\rangle _y=\pm\frac{\hbar}{2}\left|{\pm}\right\rangle _y\)
In Dirac bra-ket notation, to know what an operator does to a ket, I needs to write the ket in the basis that is the eigenstates of the operator (in order to use the eigenvalue equation.)
One way to do this to stick completeness relationships into the braket: \begin{eqnarray*} \left\langle {+}\right|\hat{S_y}\left|{+}\right\rangle = \left\langle {+}\right|(I)\hat{S_y}(I)\left|{+}\right\rangle \end{eqnarray*}
where \(I\) is the identity operator: \(I=\color{blue}{\left|{+}\right\rangle _{yy}\left\langle {+}\right|}\;+\;\color{blue}{\left|{-}\right\rangle _{yy}\left\langle {-}\right|}\). This effectively rewrite the \(\left|{+}\right\rangle \) in the \(\left|{\pm}\right\rangle _y\) basis.
Find the top row matrix elements of the operator \(\hat{S}_y\) in the \(S_z\) basis by inserting completeness relations into the brakets. (The answer is already on the Spins Reference Sheet, but I want you do demonstrate the calculation.)
face Lecture
30 min.
accessibility_new Kinesthetic
10 min.
group Small Group Activity
10 min.
spin 1/2 eigenstates quantum states
Arms Sequence for Complex Numbers and Quantum States
Students use Tinker Toys to represent each component in a two-state quantum spin system in all three standard bases (\(x\), \(y\), and \(z\)). Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT change the state of the system) and relative phase (which does change the state of the system). This activity is optional in the Arms Sequence Arms Sequence for Complex Numbers and Quantum States.