Today Quantum Mechanics (QM) is used in the university and beyond on a regular basis: Chemical bonds NMR spectroscopy The laser (blue beam in Blue-ray player; red beam in a DVD player for example) The fingerprint /signature of a molecule or atom in any spectrum-due to quantization; that is, if any energy were allowed there would be no fingerprint These days many materials have important structural features at the atomic/molecular scales; understanding their properties require QM How does it work? QM describes the microscopic world in a way analogous to how classical mechanics (CM) describes the macroscopic world.
Say we have a particle of mass m moving along the x-axis under a force F(x,t) m F(x,t) In CM to learn about the particle we describe it s position at any given time, t by x(t). One we know that, we know its momentum and kinetic energy To get x(t) we use Newton s second law: x V = potential energy. Knowing x(0) and V(0) allows x(t) to be found.
QM approaches the problem differently. Here the function are looking to describe the system is the wavefunction Ψ(x,t) of the particle. To get that we solve the Schrodinger equation: Here: and We will work a lot with this equation at various levels of sophistication.
When do we use Quantum Mechanics? (Engel 2.1) Basically, when λ is close in magnitude to the dimensions of the problem, and to the degree that the system has a discrete energy spectrum The Schrodinger Wave Equation (Engel 2.3) In QM, the behavior of a particle is described by its wave function Ψ(x,t) which we get by solving: Time-dependent SWE = 1.054573 x 10-34 J-s =Hamiltonian operator= H When V(x,t) =V(x); i.e., not a function of time Time-independent SWE
What does have the solution Ψ(x) do for me? (Engel 3.1) Born statistical interpretation of the wave function says that Ψ(x,t) 2 gives the probability of finding the particle at point x, at time, t. = probability of finding the particle between x and x+dx at time t. Low probability High probability
The Born interpretation introduces indeterminacy into QM; even if you know everything about the particle (it s wavefunction) QM can only offer statistical information about the possible results. Max Born
Wave-Particle Duality Wave-particle duality refers to the fact that both light and matter can exhibit either particle-like behavior or wave-like behavior depending on how we observe them. i.e. behavior depends on the nature of the experiment. photons can behave like particles in a photo-electric experiment electrons and other particles can exhibit a wave-like diffraction pattern 7
Heisenberg s Uncertainty Principle mid 1920s The wave-particle duality of both light and matter leads to some very awkward results. Consider the measurement of the position of an electron. If we want to measure the electron within a distance Δx we must use something of spatial resolution less than Δx. One way to achieve this is to use light of wavelength λ Δx. For us to see the electron the photon must interact with the electron. But the photon has a momentum associated with it. Thus, the very act of observing the electron leads to a change in its momentum. 8
Developing this idea fully, Werner Heisenberg showed that it is not possible to simultaneously determine the EXACT position and velocity of a particle at the same time. The greater the certainty we measure the position of a particle Δx, the less certain we can be of the particles momentum Δp (vice-versa) Heisenberg s Uncertainty Principle where ħ= h 2π h-bar The uncertainty principle is not compatible with the deterministic classical picture, since we can no longer specify exactly a particle s position and momentum simultaneously. We really can only talk about probabilities. 9
AGAIN the uncertainty principle, really only applies at the microscopic scale. h = 6.626 x 10-34 J. s e.g. The uncertainty in the position of a baseball (145 g) thrown at 90 mph (40 m/s) if we measure the momentum to a millionth of 1.0% (9x10-8 mph). x p ħ =5 x 10 35 2 (less than the radius of atomic nuclei) e.g. The uncertainty in the momentum if we locate an electron within an atom so that the uncertainty in its position is 50 pm. (Bohr radius) e - p=mv P + 10
Standard Interpretation of QM: Suppose one measures the position of a particle and finds it at location C. Where was the particle before then? The presently accepted interpretation is that it wasn`t anywhere, but the act of measurement forced the particle to take a stand : the Copenhagen interpretation. The idea (as weird as it seems) is that a particle before an observation does not have well defined properties like position and momentum. Instead the particle is viewed as being in a superposition state of many positions and momenta (with differing probabilities; particles measured in a lab are probably not close for example to Saturn). This is the real basis of the Heisenberg Uncertainty Principle. It is not a failure of not designing clever experiments. The uncertainties in x and p are part of nature!
Doesn t that bother you? It bothered Einstein a lot He famously said: [I can't accept quantum mechanics because] "I like to think the moon is there even if I am not looking at it."
Let s look at superposition states a bit more closely. Schrodinger s Cat Schrodinger thought superposition states were ridiculous. Superposition states imply particles can be in different places with different energies at the same time! To be facetious he suggested the following scenario: Place a cat in a closed box. Imagine a mechanism inside the box with cat where a vial of cyanide is released when the radioactive decay of a nucleotide triggers a Geiger counter which in turn causes a hammer to fall, breaking the vial. While radioactivity is characterized by a half life one cannot predict when any specific atom will decay. Thus within the half-life with the box closed we don t know if the cat is alive or dead.
According to QM the wavefunction of the situation can be written as: According to the Copenhagen interpretation of QM the cat is both alive and dead but by looking in the box we collapse the superposition 2 1 state to measure the cat to be dead 50% of the time or alive 2 50% of the time.
The Schrodinger Cat thought experiment which was offered as a criticism of the Copenhagen interpretation. Experiments have been done which create and probe superposition states. Bottom line: they can be generated experimentally. There has been a great deal of thought put into other interpretations of QM As only one example: Many World s interpretation Opening the box produces 2 realities: an observer who sees a live cat and an observer who see a dead cat but the two realities don't "communicate" with one another. All events lead to a bifurcation leading to an ever growing number of realities and worlds. Popular because it is simple to understand and wavefunction collapse is not invoked. However, Occam s razor suggests that it requires too many universes if true, and other universes cannot be observed.
Einstein hated the probabilistic nature of QM. He said (somewhat infamously): "As I have said so many times, God doesn't play dice with the world." He believed that QM was an incomplete theory; that perhaps there were hidden variables which if we knew them would convert QM into something deterministic. Enter David Bohm (1952) who reformulated QM in a way that was first suggested by de Broglie (so-called pilot-wave theory (1927)) but which scorned so much by Wolfgang Pauli that de Broglie never did much afterwards. Bohm In addition to a wavefunction on the space of all possible configurations, it also postulates an actual configuration that exists even when unobserved. The evolution over time of the configuration (that is, of the positions of all particles or the configuration of all fields) is defined by the wave function via a guiding equation. Pauli
So is QM correct or is there a better theory where the hidden variables whatever they may be are revealed so that before a measurement particles can actually be thought as having a definite local position and velocity?
EPR Experiment (1935) Thought experiment designed to disprove Heisenberg Uncertainty Principle and hence Quantum Mechanics "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?". Physical Review 47: 777 780 (1935).
Since we know the velocity and position of A exactly we have violated the Heisenberg uncertainty principle. Quantum Mechanics is wrong. Let A and B move far away from one another in an equal and opposite manner. Now we do an experiment where we measure the velocity (momentum) of A. Assumes properties of electrons are local and well defined At the same time do an experiment where we measure the position of particle B. Since A was moving in an equal but opposite way to B then we also exactly know the position of A.
A bombshell. So which is it: QM as we are taught or something involving hidden variables? Enter: John Bell (1964) a particle physicist at CERN who was very disturbed by QM, and worked on asking if regular QM and hidden variable QM could be distinguished experimentally. Result: Bell s Theorem, a mathematical inequality, which non-mathematically states: No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics. Put another way, it says if quantum mechanics is complete then local realism is impossible, and non-local phenomena are possible.
Bell s Theorem allowed experiments to be designed and carried out (usually involving the use of polarized light) to distinguish the two theories. Alain Aspect, Jean Dalibard, Gérard Roger (1982), "Experimental Test of Bell's Inequalities Using Time-Varying Analyzers", Phys. Rev. Lett. 49 (25): 1804 7 And the Loser is?
Einstein He was a genius but he was wrong! QM has experimentally been shown to be a complete theory. However he was wrong in such a profound way that it led to a deeper understanding of nature Sadly Einstein died in 1955 and didn t live to see the outcome of this amazing intersection of physics and philosophy. Bell died of cancer in 1990. Had he lived he surely would have been awarded the Nobel Prize.
Here is a weird thing which Einstein (AND Schrodinger) hated: The wavefunctions of the two particles in the EPR experiment at t = 0 are entangled (particle properties are no longer independent of one another). This means that a change in spin of an entangled system (whether it is up or down) will create an equal and opposite change in the partner even if they are separated by light years. This appears to violate special relativity because it appears that information is being communicated between particles faster than the speed of light. Einstein called that spooky physics at a distance.
Experiments have now been done which show quantum physics is in fact non-local (although no information can be transferred in a way that violates special relativity). Put another way non-locality is the price that is paid because particles have no definite properties before they are measured. One simple-minded way of thinking of this is that quantum objects are only probability waves before measurement but become particle-like when a measurement is made. For more can read Chapter 6.6 in Engel.
Double Slit Experiment with electrons Considered for example by Feynman to be the essence of QM https://www.youtube.com/watch?v=dfpeprq7ogc
Entanglement occurs when particles are "close to one another. Since everything is believed to have started in the Big Bang, all atoms are interconnected. This type of unity appears to be a foundation of the universe. "In the beginning there were only probabilities. The universe could only come into existence if someone observed it. It does not matter that the observers turned up several billion years later. The universe exists because we are aware of it." Cosmologist and Astrophysicist: Martin Rees
The "meaning" of Quantum Mechanics is deeply philosophical at best and completely mysterious at worse. Still as a working theory it is wildly successful. Therefore, in the words of N. David Mermin: If I were forced to sum up in one sentence what the Copenhagen interpretation says to me, it would be 'Shut up and calculate!'
Do not keep saying to yourself if you can possibly avoid it, But how can it be like that? because you will get down the drain into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that R. P. Feynman 29