Ch 4: Fundamentals of Quantum Mechanics Flashcards
What is quantum mechanics?
the most complete theory that describes the behaviour of electrons, and allows us to explain many important biological, physical, and chemical processes
What is diffraction?
the bending of waves
What is interference?
a property of any kind of wave
ie. x-ray images are the result of interference due to differential absorption of x-rays by bone tissue
What happens if x-rays pass through a periodic crystal lattice of gold atoms?
they form concentric circles, much like water waves
What suggests that electrons behave like waves?
consider a beam of electrons passing through a lattice of gold atoms:
- if one measures the number of electrons arriving at a certain point after they pass through the lattice, the result is a similar pattern of concentric circles
- this suggests that electrons can interfere constructively and destructively?
- the same experiment can be performed with neutrons
Each particle in a beam behaves as a wave.
as each wave hits an atom in the crystal, the wave splits and its different parts interfere with each other, just like a stream of water produces an interference pattern when engulfing an obstacle
A common misconception is to think that different electrons interfere with each other.
each electron must interfere with ITSELF
How can electrons and other elementary particles behave as waves?
to start, a particle can be described by its momentum
What is momentum?
a universal characteristic of any particle: mass times velocity
Why is a wave described by a wavelength?
waves generally have a repeating pattern
What is a wavelength?
distance between the nearest points of a repeating pattern
What did de Broglie hypothesize?
all particles can behave as wells
How would wave-like behaviour be detectable because Planck’s constant is so small?
detectable only when the momentum is very small
Since momentum is mass times velocity, only particles with very small masses can exhibit wave-like behaviour.
for subatomic particles, their masses are extremely small which means that they can have detectable de Broglie wavelengths even when moving with reasonable speed
What are travelling waves?
repeating and periodic disturbances that travel from one location to another
ie. waves that travel through water, sound waves that travel through air
What do waves transport?
energy, not matter
ie. water wave is the propagation of an energy burst that disturbs an otherwise still medium
What does electromagnetic radiation (including light) transmit energy through?
oscillating electric and magnetic fields, not matter
What are transverse waves?
they oscillate in a direction perpendicular to its direction of motion
What is amplitude?
difference between the mid-point and the wave crest or trough
What is wavelength?
length of one complete cycle of a wave, easily described as the distance between consecutive crests or troughs
What is the frequency of a wave?
number of complete cycles of the wave that transmit per unit time, or how “frequent” the particles in the wave vibrate as a result of the wave passing through
What are intensity and energy of a wave proportional to?
the square of its amplitude
What is a crest?
top (maximum) of the wave
What is a trough?
bottom (minimum) of the wave
What is a sine wave?
smooth repetitive oscillation
many wave phenomena in nature, including electromagnetic radiation (light), and de Broglie waves, can be represented by the sine mathematical function
What is a standing wave?
- bound on either side
- still
- do not carry energy from one location to another
- can be formed when travelling waves of equal amplitude and frequency, but moving in the opposite directions, interfere with each other
What are nodes?
points on a standing wave where the oscillation amplitude is zero
Review the one-dimensional Particle-in-a-box Model in ChIRP.
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What is another property of a wave?
an extended size
ie. water wave can occupy an entire swimming pool, the wave simultaneously exists everywhere
If an electron is represented by a wave that stretches from one end of the box to another, where is the electron?
if an electron is represented by a wave stretching over the entire Particle-in-a-box, according to Quantum mechanics, this electron exists simultaneously everywhere in the box
If an electron is represented by a wave stretching over the entire Particle-in-a-box, according to Quantum mechanics, this electron exists simultaneously everywhere in the box. How can we prove this experimentally?
- place a quantum particle in a box
- using electromagnetic fields, prepare the particle in a state described by some quantum wave
- open the box and look where the particle is
- this experiment can be repeated multiple times starting with identical conditions
- despite initial conditions being identical, every time we open the box and look, we will find the particle in a different location because the particle derives from a quantum wave
What is probability?
ie. an electron is noticed to be found more near the middle of the box
- the probability of finding the electron near the middle of the box is higher than the probability of finding the electron elsewhere
If an electron is a wave, where is it located?
according to de Broglie, a particle with precise a momentum has a precise wavelength
such an electron can be represented by an infinitely repeating wave, like a sine function, which extends to infinity, implying the particle could be located anywhere
What is the Heisenberg uncertainty principle?
one cannot know precisely (with zero uncertainty) both the position and momentum of a microscopic particle at the same time
Why is the Heisenberg uncertainty principle important?
when we need to predict the motion of objects in everyday life, we rely on the simultaneous knowledge of both momentum and location
in order to predict motion, we compute its trajectory by solving Newton’s equations
What are Newton’s equations?
give the location of a particle and its momentum, Newton’s equations predict where the particle will be at the next moment of time
What does the Heisenberg uncertainty principle tell us about Newton’s equations?
we cannot apply the equations to microscopic particles that must be treated as waves therefore, we need another fundamental theory that predicts the properties of particles that behave as waves aka QUANTUM MECHANICS
What is quantum mechanics?
theory that allows us to compute the properties of waves that describes particles
once we know the properties of waves, we can make predictions
What is classical mechanics?
assumes that the motion of particles is governed by newton’s laws
How is quantum mechanics fundamentally different from classical mechanics?
because particles are described by waves, there is no certainty, only probability
even if we know where the electron is now, we cannot predict with certainty where it will be in the next moment
what we can predict is the probability that the electron will be in a particular volume of space
What is the most important concept of quantum mechanics?
wavefunction
What is wavefunction?
a mathematical function that allows us to compute the probability to find the electron in a particular volume of space
How is wavefunction computed?
solving the Schrodinger equation
What is potential energy? (V)
ie. in a hydrogen atom, the electron is moving in the Coulomb field of the proton
- in this case, ‘V’ the potential energy of the electron from the Coulombic force between the nucleus and electron
When is the potential energy zero?
if we consider a particle that is moving in free space without any external forces
Important point of solving the Schrodinger equation:
once we find the values of the total energy, we can fix the value of E to one of those values and find the wavefunction by solving the Schrodinger equation
for each different value of E, the equation will be diffrent, so the wavefunction will be different
by repeating the calculations for each different value of E, we find all possible wavefunctions
IMPORTANT NOTE ON WAVELENGTH:
for each value of energy, there is a different wavefunction
What does each wavefunction describe?
the behaviour of the quantum particle with the corresponding value of total energy
What does the wavefunction tell us about the particle?
when we observe a quantum particle, we always see it as a particle in a well-defined location.
at the same time, we know that the particles behave as a wave (ie. it exists in multiple locations at the same time) before we observe it
THESE TWO STATEMENTS CONTRADICT EACH OTHER
How can we reconcile the two statements that describe what the wavefunction tells us about a particle?
- assume that a quantum particle always comes in and out of existence
- when it disappears in point of space, it reappears in another point of space; and these two points of space may be far apart
- when we observe the particle, we always see it as a particle because it happens to appear in a particular point of space at the moment of the observation
- if we observe it again in the next moment of time, we may find the particle in a completely different location because it will have disappeared from the original location and reappeared in the new location
How does the act of coming in and out of existence happen?
according to the magnitude of the wavefunction
instead of magnitude though, it is more convenient to define another quantity that is always positive: the square of the wavefunction
What happens when the wavefunction is greater?
the quantum particle appears more often in the areas of space where the wavefunction is greater
What does the square of the wavefunction give?
the probability of finding the particle in a particular area of space and the distribution of values of the square of the wavefunction is the probability distribution
What is a phase?
sign of the wavefunction, positive or negative
What does each different wavefunction describe?
a different state of a particle (defined by ‘n’)
What is ground state?
n=1
What is the excited state?
when the particle has an energy higher than the ground state energy
What is zero-point energy?
the difference between the ground state energy and zero
What is quantization of energy a result of?
confining the particle
whenever a quantum particle is confined its energy is quantized
When does quantum mechanical description converge to classical description?
when the particle or system becomes large (that are macroscopic objects)
this means that quantum mechanics is a complete theory accounting both for the behaviour of microscopic particles and predicting Newton’s Laws for macroscopic particles
How can particles change their energy levels?
by absorbing or emitting energy, for example from receiving energy from light or from collisions
scientists use this to study atoms and molecules, because we can measure the incident light and emitted light from particles during experiments
REVIEW OF IMPORTANT QUANTUM MECHANICS CONCEPTS (5)
- quantization arises from confinement thus the energy of a particle in a one-dimensional boxes quantized (can only take certain values)
- energy of a particle in a one-dimensional box is dependent on the integer number ‘n’, the mass ‘m’, and the size of the box ‘L’
- wavefunction and energy states are labelled by quantum numbers, which foe the particle in a box are the positive integers (n=1, 2, 3, …)
- lowest energy possible for a particle in a one-dimensional box corresponds to the ground state, n=1
- wavefunction and the probability distribution each have n-1 nodes
