Students use their arms to act out two spin-1/2 quantum states and their inner product.

Basic algebraic and geometric properties of the dot product.

In this small group activity, students draw components of a vector in Cartesian and polar bases. Students then write the components of the vector in these bases as both dot products with unit vectors and as bra/kets with basis bras.

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.

For this problem, use the vectors \(|a\rangle = 4 |1\rangle - 3 |2\rangle\) and \(|b\rangle = -i |1\rangle + |2\rangle\).

1. Find \(\langle a | b \rangle\) and \(\langle b | a \rangle\). Discuss how these two inner products are related to each other.
2. For \(\hat{Q}\doteq \begin{pmatrix} 2 & i \\ -i & -2 \end{pmatrix} \), calculate \(\langle1|\hat{Q}|2\rangle\), \(\langle2|\hat{Q}|1\rangle\), \(\langle a|\hat{Q}| b \rangle\) and \(\langle b|\hat{Q}|a \rangle\).
3. What kind of mathematical object is \(|a\rangle\langle b|\)? What is the result if you multiply a ket (for example, \(| a\rangle\) or \(|1\rangle\)) by this expression? What if you multiply this expression by a bra?

The properties that an inner product on an abstract vector space must satisfy can be found in: Definition and Properties of an Inner Product. Definition: The inner product for any two vectors in the vector space of periodic functions with a given period (let's pick \(2\pi\) for simplicity) is given by: \[\left\langle {f}\middle|{g}\right\rangle =\int_0^{2\pi} f^*(x)\, g(x)\, dx\]

1. Show that the first property of inner products \[\left\langle {f}\middle|{g}\right\rangle =\left\langle {g}\middle|{f}\right\rangle ^*\] is satisfied for this definition.
2. Show that the second property of inner products \[\left\langle {f}\right|\Big(\lambda\left|{g}\right\rangle + \mu \left|{h}\right\rangle \Big) = \lambda\left\langle {f}\middle|{g}\right\rangle +\mu\left\langle {f}\middle|{h}\right\rangle \] is satisfied for this definition.

This small group activity is designed to help students visualize the cross product. Students work in small groups to determine the area of a triangle in space. The whole class wrap-up discussion emphasizes the geometric interpretation of the cross product.

Part I

Using the applet at Inner Products of Functions:

1. Sketch the function \(f(x)g(x)\) for the default functions.
2. Make a list of properties of the functions \(f(x)\) and \(g(x)\) that you used to make your sketch. Pay attention to special cases and symmetries.
3. Check your sketch by clicking on the green check box, i.e. refer to authority.
4. Plug other functions into the applet and check that your list of properties is accurate and complete, i.e. check many cases.

Part II

Use the applet to find the values of the following integrals for integer \(m\) and \(m'\): \begin{align*} &\int_0^{2\pi} \sin mx\;\sin m'x \;dx\\[12pt] & \int_0^{2\pi} \sin mx\;\cos m'x \;dx\\[12pt] &\int_0^{2\pi} \cos mx\;\cos m'x \;dx \end{align*}

Part III

Do a simple change of variables in your integrals to convince yourself of the following:

For integer \(m\) and \(m'\) \begin{align*} &\int_0^L \sin\tfrac{2\pi mx}{L}\;\sin\tfrac{2\pi m'x}{L} \;dx= \begin{cases}\frac{L}{2} \mbox{ if } m = m'\\ 0 \mbox{ if } m \neq m'\end{cases}\\[12pt] & \int_0^L \sin\tfrac{2\pi mx}{L}\;\cos\tfrac{2\pi m'x}{L} \;dx= \begin{cases}0 \mbox{ if } m= m'\\ 0 \mbox{ if } m \neq m'\end{cases}\\[12pt] &\int_0^L \cos\tfrac{2\pi mx}{L}\;\cos\tfrac{2\pi m'x}{L} \;dx= \begin{cases}\frac{L}{2} \mbox{ if } m= m'\neq 0\\ L \mbox{ if } m=m'=0\\0 \mbox{ if } m \neq m'\end{cases}\\[12pt] \end{align*} Hint: Recall that the function transformation \(f(x)\rightarrow f(\alpha x)\) shrinks or expands the function \(f(x)\) along the \(x\)-axis. See GMM: Function Transformations

Part IV

Compare your results for sines and cosines integrated over a whole period (this example) to what you know about the energy eigenstates of a quantum infinite square well, i.e. compare to a known example. How are these examples the same or different? Can you use the applet to explore the infinite square well case?

A short improvisational role-playing skit based on the Star Trek series in which students explore the definition and notation for position vectors, the importance of choosing an origin, and the geometric nature of the distance formula. \[\vert\vec{r}-\vec{r}^\prime\vert=\sqrt{(x-x^\prime)^2+(y-y^\prime)^2-(z-z^\prime)^2}\]

• Outer products yield projection operators
• Projection operators are idempotes (they square to themselves)
• A complete set of outer products of an orthonormal basis is the identity (a completeness relation)

Students learn how to express Angular Momentum as a vector quantity in polar coordinates, and then in Cylindrical and Spherical Coordinates

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Students take the inner product of vectors that lie on the spacetime axis to show that they are orthogonal. To do the inner product, students much use the Minkowski metric.

• Recall the relationship between the sign of the dot product and the orientation of the vectors.
• Use graphical methods to estimate the value of a vector line integral.
• Lay the groundwork for thinking about conservative and non-conservative vector fields.