Waves And Quantum Waves 3 Notes

Notes for Waves and Quantum Waves, Part 3

Quantum-mechanical wavefunctions:

The de Broglie hypothesis states that particles can be thought of as matter waves of wavelength λ=h/p. They can therefore be described by a wavefunction ψ(x,t) of the general form:

ψ(x,t)=ψ₀e^i(ωtkx)

Using the de Broglie equation, constants k and ω can be written as:

Thus, the wavefunction for a mono-energetic particle travelling in the positive x direction with energy E and momentum p is:

$$\boxed{\mathbf{\psi}\left( \mathbf{x,t} \right)\mathbf{=}\mathbf{\psi}_{\mathbf{0}}\mathbf{e}^{\frac{\mathbf{i}}{\mathbf{\hslash}}\left( \mathbf{Et - px} \right)}}$$

Physical meaning of the wavefunction:

The wavefunction ψ itself has no physical meaning, however |ψ|² is the probability density function for the particle described by ψ, such that:

Is the probability of finding the particle in a line element dx, area element dA=dxdy or volume element dV=dxdydz centred on (x,y,z), with normalisation condition:

That is, the particle must be found somewhere in space at a given time.

This physical meaning of ψ can be understood by considering:

1. Energy of a classical wave is E|ψ|²

2. For a monochromatic light beam, energy is quantised into photons of equal energy

3. For a light beam EN, where N is number of photons per unit volume

4. Therefore |ψ|²N

5. In some volume element dV, the probability of finding a photon is N

6. Hence |ψ|² represents the probability of a photon being observed in dV

7. By de Broglie hypothesis, particles exhibit both particle and wave behaviour, like photons

8. Hence |ψ|² represents the probability of a particle being observed in dV

Expectation value of position x:

In the time-independent, one-dimensional case, particle position x is described by the probability density function |ψ(x)|²=ψ^*(x)ψ(x). Any one measurement of x will be an inaccurate description of the nature of the particle, which is in fact described by a probability density distribution.

The best single value describing the ‘true position’ of the particle to greatest accuracy is therefore the expectation value ⟨x⟩:

⟨x⟩=xψ^*ψdx

The one-dimensional, time-independent Schrödinger equation:

The one-dimensional, time-independent Schrödinger equation describes the behaviour of a mono-energetic particle with energy E travelling along a line with some position x. The particle has a wavefunction $\psi(x,t) = \psi_{0}e\hat{}(i/\hslash(Et - px))$ and a potential V(x) which varies with x:

$$\widehat{\mathbf{T}}\mathbf{\psi}\left( \mathbf{x} \right)\mathbf{+}\widehat{\mathbf{V}}\mathbf{\psi}\left( \mathbf{x} \right)\mathbf{= E\psi(x)}$$

Where T is the kinetic energy operator, V is the potential energy operator, and E is the total energy of the particle:

$$\mathbf{\therefore}\boxed{\mathbf{-}\frac{\mathbf{\hslash}^{\mathbf{2}}}{\mathbf{2}\mathbf{m}}\frac{\mathbf{d}^{\mathbf{2}}\mathbf{\psi}}{\mathbf{d}\mathbf{x}^{\mathbf{2}}}\mathbf{+}\mathbf{V(x)\psi}\mathbf{= E\psi}}\mathbf{\ or}\mathbf{\ }\boxed{\frac{\mathbf{d}^{\mathbf{2}}\mathbf{\psi}}{\mathbf{d}\mathbf{x}^{\mathbf{2}}}\mathbf{+}\mathbf{k}^{\mathbf{2}}\mathbf{\psi =}\mathbf{0}}\mathbf{,\ \ k =}\frac{\sqrt{\mathbf{2}\mathbf{m}\left( \mathbf{E - V} \right)}}{\mathbf{\hslash}}$$

The Schrödinger equation can be shown to obey the wave equation and hence describe the wavelike properties of the particle:

$$\mathbf{\therefore}\frac{\mathbf{\partial}^{\mathbf{2}}\mathbf{\psi}}{\mathbf{\partial}\mathbf{x}^{\mathbf{2}}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{c}^{\mathbf{2}}}\frac{\mathbf{\partial}^{\mathbf{2}}\mathbf{\psi}}{\mathbf{\partial}\mathbf{t}^{\mathbf{2}}}$$

Where $k = \sqrt{2m(E - V)}/\hslash$, or $\sqrt{2m(E)}/\hslash$ for a zero potential function. Generalising to two and three dimensions, the wavevector k=(k_x,k_y,k_z) is given by:

$$\left| \mathbf{k} \right|^{2}\mathbf{=}\frac{2m\left( E - V(x) \right)}{\hslash^{2}}$$

Conservation of energy for the Schrödinger equation:

Energy is conserved for particles which obey the Schrödinger equation:

$$\psi E = V\psi - \frac{\hslash^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}},\ \ \psi = \psi_{0}e^{\frac{i}{\hslash}\left( (E - V)t - px \right)}$$

Substituting the wavefunction into the equation:

$$\psi E = \frac{\hslash^{2}}{2m}\left( \frac{p}{\hslash} \right)^{2}\psi + V\psi = \psi\left( \frac{p^{2}}{2m} + V \right)$$

$$\mathbf{\therefore E =}\frac{\mathbf{p}^{\mathbf{2}}}{\mathbf{2}\mathbf{m}}\mathbf{+}\mathbf{V}$$

Where p²/2m gives the kinetic energy and V gives the potential energy.

Particles in infinite-potential wells:

An infinite potential well totally confines a particle such that the Schrödinger equation only has a solution inside the well:

V(x)=0,0<x<a

V(x)=,x<0,x>a

Boundary...

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