quantum physics Flashcards
What are de broglie waves?
De Broglie said that if photons are particles with wave like properties then why couldn’t other particles act in the same way.|He suggested the wavelength of particles could be described using |λ=h/p=h/γmv |for which we usually use to describe a photon
How did De Broglie come to this relation?
we know from the photoelectric effect and blackbody radiation that E=hv and we defined E=pc for a massless particle travelling at the speed of light…|we can then equate the relationship…|p=hv/c|and thus we can say |λ=h/p||De broglie suggested we could use this relation to describe particles of mass and not just photons.
How was the De Broglie hypothesis proved?
if we look at the wavelength of an electron moving at 10⁷ m/s (v<
What is the wavefunction, .ψ, of an particle?
the wavefunction, which has a value at every point in space and time, is related to the probability of finding the particle at that point in space and time. ||ψ is the amplitude of the wave associated with a particle, it can have both positive and negative values. |ψ itself is not the probability and is not directly observable from experiments|the wavefunction is often expressed as a complex quantity even though physical observations can only be real.
How do we define the absolute magnitude of our wavefucntion ψ?
as ψ can be a complex quantity it can be written as |ψ=a+ib ||ψ|=|a+ib|=√(a²+b²)||ψ|²=ψψ*| =a²+b²||we have defined the complex conjugate ψ=a-ib to find the absolute magnitude of ψ.||ψ|² is useful for when we start to deal with complex numbers.
How is |ψ|² used to defined the probability of finding a particle?
if we have a small volume around some point (dx in one dimension and dv in 3 dimensions), then the probability of finding the particle in that volume is given by…|1D: P(x) dx = |ψ(x)|² dx|3D: P(x,y,z) dv = |ψ(x,y,z)|² dv||note in one dimension, its the probability of locating its x coordinate, not the exact location of the particle.
What are the differences between the quantum and classical interpretations of the wavefunction?
in classical physics, the wavefunction is zero everywhere in space except at a single point where it would have a nonzero value. |this point corresponds to the classical position of the particle.||in quantum theory we can have any type of wavefunction we like, |as we spread out the weight of the wavefunction over space, we also distribute the probability of finding that particle.|it no longer makes sense to talk about the position of the particle as the particle can be found in a range of positions
What are the consequences of the new quantum definition of the wavefunction?
- we shouldnt talk about the position of a particle |2. if we look for a particle, we will always find that it is either there or not there, and since the particle must always be somewhere, we know that in one dimension -∞
What does the wavefunction allow us to calculate ?
- the probability of finding a particle |2. the average value of other physical quantities called expectation values. |for an arbitrary function f(x), its expectation value is determined by |||once we know the wavefunction of a system we can determine the relevant parameters that describe it
How can we then model a particle using the wavefunction?
- consider a wave with wavelength λ and wavevector k=2π/k so that |ψ(x)=Asin(kx) |this wave has a perfectly defined wavevector K and therefore momentum p.|however the probability of locating (|ψ|²) is spread equally over many different locations so it is fully delocalised, meaning it has no clearly defined position. |2. to change this, we use the principle of superposition. If we add another component tot he wave with a different wavelength then there will be both constructive/destructive interference.|3. the particle will then be more likely to be in some places than other places |4. the more wavevectors/wavelengths we superimpose, the more localisation we can make the particle have. |5. allows us to create a wavepacket (group of superimposed waves which together form a travelling localised disturbance)|THIS MODEL IS WHAT WE REGARD AS REPRESENTING A PARTICLE
What are the limitations of the model of the particle?
by adding more components to the wave with different wavevectors, we can more clearly define the location of a particle but the overall wavevector or momentum becomes less clearly defined||therefore if we think of a particle as having a wavefunction that describes its probability, then there is a fundamental limit on how precisely we can measure quantities like position.
How do these limitations lead onto Heisenburg’s uncertainty principle?
first case: many wavelengths/wavevectors superimposed|range of values we’d expect to find x is narrow so uncertainty in position is small but the wavelength is not well defined as not many oscillations so ∆p large.||second case: less wavelengths/wavevectors superimposed |the wavelength is well defined so ∆p small but the position is not well defined so ∆x large||we can see there is a trade off between ∆x and ∆p. After mathematical analysis we see that the best we can do is the shape of a bell curve in terms of minimising both where…|∆x∆p=h/4π||For all other wavefunctions then, the product of ∆x∆p≥h/4π (=hbar/2)|(h(bar)=h/2π)|this is Heisenberg’s uncertainty principle ||the uncertainty in position and momentum arises because of oscillations in the wavefunction of a particle in space.
How the uncertainty principle relates to measurements we take?
∆x∆p≥h/4π |if we assume the electron is moving slowly enough to ignore relativistic corrections. |if we measure electron with uncertainty 1µm, ∆x=1µm|∆p≥h/4π∆x|∆v≥h/(4mπ∆x)|so after a period of one second, the elctron could have moved |∆x≥h/(4mπ∆x)×1 second|which here we work out to be 58m if initial uncertainty in measurement is 1µm.
What happens if we consider the oscillations of the wavefunction in time?
ψ(x,t)=Asin(kx-ωt) |E=hv=h(bar)ω|ans the uncertainty principle tells us that |∆E∆t≥h(bar)/2||t is the time that a particle will stay in a certain state before switching to another state. This means that a particle can only have a perfectly defined energy if stays that way forever (doesnt interact with anything) |if it does interact with something this causes a change in its state after a certain time, then there must be some uncertainty in its energy.||a consequence of this is that we are allowed to violate the conservation of energy, if we do so for short amounts of time.
Describe the particle in a box/quantum well model
An important consequence of quantum theory is the quantisation of the energy of a particle. |-|1. consider a particle in free space|-if there is nothing to interfere with the particles motion, then it has no restriction on its motion.||2. Now, we put this particle in a box with infinitely hard walls |-we consider this problem in one dimension for simplicity |-the problem is also called the quantum well due to shape of the potential looking like an infinitely deep well. ||3. The potential between x=0 and x=L is 0, while it is infinite everywhere else |-since the particle being in area of infinite density would require it to have infinite energy, ψ must be equal to zero everywhere except between 0 and L.|(set of solutions is the same as that of a string stretched between two fixed points) ||4. If we analyse the mathematics of this situation we find that when you confine a particle to a finite space, the energy levels are quantised,|- this is in contrast for a free particle for which the K.E spectrum is continuous
