1 / 9

V(x)=0 for L>x>0 V(x)=∞ for x≥L, x≤0

Region II. Region III. Region I. V(x)=0. V(x)= ∞. V(x)= ∞. L. 0. x. Particle in a 1-Dimensional Box. Time Dependent Schrödinger Equation. PE. KE. TE. Wave function is dependent on time and position function:. 1. Time Independent Schrödinger Equation. V(x)=0 for L>x>0

maris
Download Presentation

V(x)=0 for L>x>0 V(x)=∞ for x≥L, x≤0

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Region II Region III Region I V(x)=0 V(x)=∞ V(x)=∞ L 0 x Particle in a 1-Dimensional Box Time Dependent Schrödinger Equation PE KE TE Wave function is dependent on time and position function: 1 Time Independent Schrödinger Equation V(x)=0 for L>x>0 V(x)=∞ for x≥L, x≤0 Applying boundary conditions: Classical Physics: The particle can exist anywhere in the box and follow a path in accordance to Newton’s Laws. Quantum Physics: The particle is expressed by a wave function and there are certain areas more likely to contain the particle within the box. Region I and III: Region II:

  2. Finding the Wave Function Our new wave function: But what is ‘A’? This is similar to the general differential equation: Normalizing wave function: So we can start applying boundary conditions: x=0 ψ=0 x=L ψ=0 where n= * Calculating Energy Levels: Since n= * Our normalized wave function is:

  3. Particle in a 1-Dimensional Box Applying the Born Interpretation n=4 n=4 n=3 E n=3 E n=2 n=2 n=1 n=1 x/L x/L

  4. Particle in a 2-Dimensional Box Doing the same thing do these differential equations that we did in one dimension we get: A similar argument can be made: In one dimension we needed only one ‘n’ But in two dimensions we need an ‘n’ for the x and y component. Lots of Boring Math Our Wave Equations: Since For energy levels:

  5. v E u w u w 0 a C3 u w v σ2 σ1 v u v w σ2 w u v Particle in a 2-Dimensional Equilateral Triangle Let’s apply some Boundary Conditions: Types of Symmetry: A C23 Defining some more variables: w v u So our new coordinate system: Our 2-Dimensional Schrödinger Equation: Solution: Substituting in our definitions of x and y in terms of u and v gives: Where p and q are our nx and ny variables from the 2-D box!

  6. Finding the Wave Function Substituting gives: So what is the wave equation? It can be generated from a super position of all of the symmetry operations! So if: And we recall our original definitions: But what plugs into these? If you recall: Substituting and simplifying gives: A1 Continuing with the others: A2 Energy Levels:

  7. Plotting in Mathematica p=1 q=0

  8. A1

  9. A2

More Related