8. Quantum Mechanics basics Flashcards
What is the equation for the wavelength of a particle’s wavefunction?
λ = h / p
h- plank constant
p-momentum
What determines the momentum of a particle?
The wavelength of its wavefunction, with a longer wavelength implying smaller momentum
What numerical form describes the wavefunction?
It is a continuous, complex and single-valued function of space coordinates and time
In terms of the wavefunction, if a particle is moving through space with a known momentum, why is the position unknown?
Includes an intuitive explanation of Heisenberg’s uncertainty principle in terms of the particle’s wavefunction.
The position is unknown due to the fact that when the momentum is known, the superposition of possible wavefunctions collapses down to one wavelength, and now the wavefunction has the same amplitude (it is a complex number that spirals around the axis with constant amplitude) at all points in space.
We can have some knowledge of the particle’s location if the wave-function consists of the sum of several different waveforms with different wavelengths since in this case, the amplitude of the particle’s wavefunction is different in different positions. However, this means we have less knowledge of the particle’s momentum an the wavefunction is composed of waveforms of different wavelengths (wavelength gives momentum)
In terms of the wavefunction, how do you find the probability of measuring the particle at a given location?
The probability is given by the square of the amplitude of the wavefunction.
What is the equation for the energy of a wavefunction?
h(bar) * ω(frequency)
What is h(bar)?
plank constant h divided by 2π (knows as the quantised version of Plank’s constant)
What is the Schrödinger equation?
The Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system.
What is the equation for the energy of a photon in terms of plank’s constant?
E = h(bar) * ω
or
E = h * f
What is the point in confining a particle to an infinite potential well?
When a particle is traveling freely through space, there is no limitation to the wave function’s possible wavelength (and thus frequency). However, if it is placed inside a potential well (can be thought of as a box), the only wavelengths and frequencies that are possible are the ones that ensure that the amplitude of the wavefunction is close to 0 at the boundary of the box. Therefore, for a trapped particle, wavelengths between these specific frequencies with nodes at the edge of the well, are not allowed and hence only certain energy levels are possible.
What is the general solution form for the time-independent Schrodinger equation?
ψ(x) = Acos(kx) + Bsin(kx)
What is the equation for Heisenberg’s uncertainty principle?
ΔxΔpx >= h(bar)/2
Δx- uncertainty in x position
Δpx- uncertainty in x momentum
h(bar)- planks constant / 2pi
How do you derive the energy levels of a particle in a potential well?
- Modify the Schrodinger equation to the case of the infinite potential well, where the potential V is 0, this will allow you to rearrange for an expression for the 2nd time derivative of the wave function.
- Solve the Schrodinger equation using the general solution ψ(x) = Acos(kx) + Bsin(kx), plugging in the boundary conditions that the wave function is =0 at the edges of the potential well. This will give you a solution of the form ψ(x) = Bsin(kx) as the cos terms fall away, then find k using other boundary conditions, and find B using the knowledge that the integral between +/- inf of the wavefunction = 1. This will give you the equations for all the wavefunctions.
To relate the wavefunction equation to the energy of each energy level, we differentiate the wavefunction equation twice to get it to the form of the Schrodinger equation, then we can see from the coefficients an equation for E in terms of k (from sin(kx)). Then the value of k calculated earlier from the boundary conditions can be plugged in for a final equation for Ein terms of n, the energy level.
How can you get the kinetic energy of a particle in a certain width potential well using the uncertainty in momentum (Heisenberg)?
The uncertainty in momentum can give the momentum of particles in the lowest energy level, and thus by setting E=1/2 mv^2 (the particle just has KE, no potential at the bottom of the well), you can rearrange for E in terms of momentum p and then plug in this uncertainty in momentum to get the min KE. To get the uncertainty in momentum, the uncertainty in position is equal to half the width of the potential well.
What is the uncertainty in x used to calculate the min KE in a potential well of width a?
a
As the uncertainty principle states that the minimum uncertainty in momentum will occur when there is maximum uncertainty in position, and the max uncertainty in position is the width of the potential well.
