1. particle in a box
Potential problem

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

3. 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:

4. 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:

5. 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

6. 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:

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

8. 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

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

10. energies of confined states

11. Energies of confined states

12. 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……………

13. 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:

14. interpretation of the wave function

15. 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 :

16. normalisation of wave functions

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

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

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