12 Separation of Variables - numerical answer obtained from...

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

View Full Document Right Arrow Icon
Fundamentals of Fundamentals of Nanoelectronics Nanoelectronics Prof. Supriyo Datta ECE 453 Purdue University Network for Computational Nanotechnology 09.22.2004 Lecture 12: Separation of Variables Ref. Chapter 2.2 & 2.3
Background image of page 1

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

View Full DocumentRight Arrow Icon
Schrödinger equation (Matrix Form) 00:07 • Remember time independent Schrödinger equation: • In one dimension (say in x) if we neglect the potential U, we’ll have: • But is there any difference between the solutions of the matrix equation and the analytical solution? In particular how do the eigenvalues of the two methods compare to each other? • The answer depends on the range of energy which is considered. For low energies the two answers match well whereas for high energies the two deviate from each other and of course it is the
Background image of page 2
Background image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: numerical answer obtained from the matrix on the left that is not quit right. Let’s discuss this more in depth: (See next page) Φ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ∇ − = Φ U m E 2 2 2 = Φ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = Φ 2 2 2 2 dx d m E = • Using finite difference method, we have this matrix equation: ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − − = ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ N N t t t t t t t t t E φ . . . 2 2 2 . . . 2 1 2 1 " " " % # # # % # # # % # # # " " " " " " 2 2 2 / ma t = ≡ 1 2 3 n-1 n n+1 N a...
View Full Document

This note was uploaded on 01/12/2011 for the course ME 305 taught by Professor Wright,j during the Spring '10 term at Birla Institute of Technology & Science, Pilani - Hyderabad.

Page1 / 3

12 Separation of Variables - numerical answer obtained from...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online