2023-10-27 Quantum Physics Lecture 17

physqp1 Review 2023-10-24 Quantum Physics Lecture 16

• Quantum Tunneling
• Step Potentials and Scattering
• Bound States including Infinite square well, Finite Square Well and Quantum Harmonic Oscillator.
• Solving the Time independent Schrodinger equation for $\hat{H}$

Quantum Postulates

1. Every measurable quantity “a” is associated with an operator $\hat{A}$, that is also called an “observable”, that is related through: $$ \hat{A}\varphi(x) = a \varphi(x)

2. Any _measurement_ of $A$ gives a corresponding eigenvalue $a$ and a matching eigenstates $\varphi_{n}$ 3. $\Psi_{(x,t)}$ contains all the information of the system 1. $|\Psi|^{2}$ is the probability density, and $<A> = \int \Psi ^* \hat{A} \Psi \, dx$ gives the expectation value of $A$. 4. Time dependent $\Psi(x,t)$ obeys S.E: $$ \frac{\hat{p}^{2}}{2m} \Psi_{(x,t)} + V(x) \Psi_{x,t} = i\hbar \frac{d}{dt}\Psi_{x,t}

Hermitian Operator

If an operator has

1. Real Eigenvalues
2. Orthogonal Eigenfunctions and distinct Eigenvalues for these Eigenfunctions,
3. Eigenfunctions that form a complete set: i.e can express any function as a superposition of its constituent eigenfunctions. Then the operator is known as a hermitian operator. N.B: Observational operators are hermitian.

Example:

$${\int }_{-\infty }^{\infty }{\phi }^{\ast }(A\Psi )dx={\int }_{-\infty }^{\infty }(A\phi {)}^{\ast }\Psi dx={\int }_{-\infty }^{\infty }{\phi }^{\ast }{A}^{†}\Psi dx$$

Matrix Mechanics and Representation