2023-10-03 Quantum Physics Lecture 10

Last class 2023-09-29 Quantum Physics Lecture 9

• Solving Time independent Schrodinger equation with Separation of variables.
• Infinite square well and how to solve for boundry conditions
• Quantization of energy into discrete levels, zero point energy and ${\Psi }^{\ast }\Psi$ is time independent

superposition

The wave function given in last class is a specific example of superposition that can be generalized to the function given below.

$$\Psi (x)=c_1{\psi }_1(x)+c_2{\psi }_2(x)$$

Where $c_1$ and $c_2$ are complex numbers. We can further generalize this above function to the below function:

$$\Psi (x,t)=\sum c_n{\phi }_n(x)e^{-i{E}_{n}t/\hslash }$$

where we get

$$\phi (x)e^{-i{E}_{n}t/\hslash }={\psi }_n(x)$$

through time evolution. (adding the $e^{-i{E}_{n}t/\hslash }$ term)

Setting t=0, we can see the following

$${\phi }_m^{\ast }\Psi (x,0)=\sum C_n{\phi }_m^{\ast }{\phi }_n(x)$$ $$\int {\phi }_m^{\ast }\Psi (x,0)dx=\sum C_n\int {\phi }_m^{\ast }{\phi }_n(x)dx$$

From the integral on the the RHS of the above eq:

\int \varphi_{m}^{*}\varphi_{n} , dx

>$$ \int \sin\left( \frac{m\pi x}{L} \right) \sin\left( \frac{n\pi x}{L} \right)\, dx

This yields a situation where the function is 0 where $m\ne 0$ and 1 where $m=n$. This is known as the Kronecker delta, and is denoted by

\delta_{m,n}

\int \varphi_{m}^{*}\Psi(x,0) , dx = \sum C_{n}\delta_{m,n}=C_{m}

This above equation is equivalent to the Fourier coefficient. **NB**: $|C_{m}|^{2}\sim$ probability of a particle being in the "nth" state. For $t>0$,

\Psi(x,t) = \sum_{n} c_{n}\varphi_{n}(x)e^{-iE_{n}t/\hbar}

$$Weknowthat$$

\int ^{L} _{0} \Psi^{*}(x,t)\Psi(x,t) , dt = 1

Because the particle has to be *somewhere*. If we substitute in $\sum_{n} c_{n}^{*}\varphi_{n}^{*}(x)$ for $\Psi^{*}(x,t)$ we can derive the formula above to

\sum_{m} |C_{m}|^{2} = 1

where $C_{m}$ is the probability of finding the particle in the "m"th state. # [[Functional Vector Space]] Using the following equation,

\Psi(x,t) = \sum c_{n} \psi_{n} (x)

We can draw an analogy to geometrical vector space $\mathcal{V}$, where we have vectors $\vec{A}, \vec{B}$ that are composed of unit vectors $\hat{i}, \hat{j}\ \& \ \hat{k}$. If we take an analogy to wave-function vector space, we can compose objects (vectors), $\Psi(x,t)$ with unit vectors $\varphi_{n}$ and $\varphi_{m}^{*}$. ![[Pasted image 20231026222121.png]]' vs ![[Pasted image 20231026222135.png]] Applying the above, we can have a basis vector:

\varphi_{n}(x) = \sqrt{ \frac{2}{L} } \sin\left( \frac{n\pi x}{L} \right)

Represent an [[eigenfunction]] of the Hamiltonian operator $\hat{H}$ in the following equation:

\left[ -\frac{\hbar^{2}}{2m} \frac{d^{2}}{dx^{2}} + V(x) \right]\varphi(x) = E\varphi(x)

where the Hamiltonian operator $\hat{H}$ is represented in the square brackets. If you solve for appropriate values of E and $\varphi$, you can graph it: ![[Pasted image 20231026222619.png]]