Lecture 4 Notes Quantum Mechanics

# Lecture 4 Notes Quantum Mechanics - Lecture 4 Notes Quantum...

• Notes
• 5

This preview shows pages 1–3. Sign up to view the full content.

Lecture 4 Notes: Quantum Mechanics First Postulate of Quantum Mechanics: Any quantum mechanical particle or a system in general can be described by a wavefunction   r , t . (Depending on a system may be in a vector form). In general we will represent wavefunction in a form of a ket : r , t . The wavefunctions belong to a state space, which has the properties of a vector space. The general properties of wavefunctions are: (a) Single valued (b) Square integrable (c) Nowhere infinite (at infinity as well as elsewhere) (d) Continuous (e) Piecewise continuous first derivative Now let s consider the simplest wavefunction for a particle. In a free space such as vacuum with no external forces the particle can be approximated as a plane wave: e ikx i t , where k p is the wavevector. Since there is no external forces then the energy of a particle is its kinetic energy: E mv 2 p 2 2 k 2 . 2 2 m 2 m How does this wave propagate in time and space? To answer this question let s consider the change in wavefunction with respect to time and position: e ikx i t ike ikx i t x e ikx i t   i e ikx i t 2 e ikx i t   k 2 e ikx i t t x 2 e ikx i t   i E e ikx i t 2 p 2 e ikx i t   e ikx i t t x 2 2 2 2 p 2 i e ikx i t Ee ikx i t ikx i t ikx i t t e e 2 m x 2 2 m  E This exercise results in a curious statement about the propagation of particle ~ plane wave in free space: 2 2 e ikx i t i e ikx i t 2 m x 2 t In general in the presence of forces in 3D space energy of a particle would be:

This preview has intentionally blurred sections. Sign up to view the full version.

E V x , t , which would yield a following equation for the particle described by a wavefunction   : 2 2 r , t V x , t x , t i x , t 2 m t ______________________________________________________________________________
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern