particle in a box Potential problem

Wave functions Significance of wave function Normalisation The time-independent Schrodinger Equation Solutions of the T.I.S.E

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

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:

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

de Broglie Hypothesis In 1924, de Broglie suggested that if waves of wavelength λ were associated with particles of momentum p=h/λ, then it should also work the other way round……. A particle of mass m, moving with velocity v has momentum p given by:

Kinetic Energy of particle If the de Broglie hypothesis is correct, then a stream of classical particles should show evidence of wave-like characteristics……………………………………………

Standing de Broglie waves Eg electron in a “box” (infinite potential well) V=0 V= V=0 V= Electron “rattles” to and fro Standing wave formed

wavelengths of confined states In general, k =nπ/L, n= number of antinodes in standing wave

energies of confined states

Energies of confined states

particle in a box: wave functions From Lecture 4, standing wave on a string has form: Our particle in a box wave functions represent STATIONARY (time independent) states, so we write: A is a constant, to be determined……………

interpretation of the wave function The wave function of a particle is related to the probability density for finding the particle in a given region of space: Probability of finding particle between x and x + dx: Probability of finding particle somewhere = 1, so we have the NORMALISATION CONDITION for the wave function:

interpretation of the wave function

Interpretation of the wave function Normalisation condition allows unknown constants in the wave function to be determined. For our particle in a box we have WF: Since, in this case the particle is confined by INFINITE potential barriers, we know particle must be located between x=0 and x=L →Normalisation condition reduces to :

normalisation of wave functions

Solving the SE :an infinite potential well So, for 0<x<L, the time independent SE reduces to: General Solution:

Boundary condition: ψ(x) = 0 when x=0:→B=0 Boundary condition: ψ(x) = 0 when x=L:

In agreement with the “fitting waves in boxes” treatment earlier………………..