Science One Physics Lecture 8. The Schrodinger Equation slides with commentary

Transcription

1 Science One Physics Lecture 8 The Schrodinger Equation slides with commentary :

2 Outline The Schrödinger equation Measurement in quantum mechanics: from the Stern-Gerlach experiment to the Born rule Entanglement Quantum cryptography

3 Outline - basic QM concepts Particle-wave duality Superposition principle Heisenberg uncertainty relation The Schrödinger equation Tunneling Bound states Measurement in quantum mechanics Entanglement

4 Recap: Planck s quantum hypothesis (1900) Light caries energy in indivisible packets, or quanta. For light of frequency f, each quantum has an energy of E = hf

5 Resolution of the UV catastrophe Ultraviolet catastrophe Intensity per wavelength Planck s law (new explanation) Rayleigh-Jeans law (classical explanation) wavelength λ, μm

6 Quanta are real: the photoelectric effect f Einstein s explanation (1905): Quanta of light transfer their energy E = hf to individual electrons, kicking them out. Confirmation of Planck s quantum hypothesis

7 DeBroglie: Matter can behave as a wave (1924) p = h λ Particle-wave duality

8 Davisson-Germer experiment (1927) Same as with photon scattering! Peaks of electron intensity at distinct angles Nickel Experimental confirmation of the wave nature of particles

9 >>> End of Review <<<

10 The Schrödinger Equation (1926) Significance: Non-relativistic QM rests on two formulas. This is one. Limitations: Does not include the effects of special relativity. Does not describe measurement.

11 Deciphering...??

12 h vs. The Schrödinger Equation contains rather than Planck s constant h. The distinction between h and is a matter of convention: = h 2π. This comes from E = hf = ω, where ω = 2πf. E.g., for a sinusodial wave ψ(t), ψ(t) sin(ωt) = sin(2πf t).

13 Math interlude: complex numbers The imaginary unit i is defined as i := 1 And hence we have the relation i 2 = 1. For any pair a, b of real numbers, is a complex number. c = a + i b Therein, a is called the real part and b the imaginary part of c. Complex numbers can be added, multiplied and divided by.

14 Math interlude: complex numbers i a: real part c b:imaginary part -1 1 c = a + ib -i e iφ = cos φ + i sin φ

15 Math interlude: complex numbers Elementary rules for operating with complex numbers: (a + ib) + (a + ib ) = (a + a ) + i(b + b ) (a + ib)(a + ib ) = (aa bb ) + i(ab + ba ) (a + ib) 1 = a ib a 2 + b 2 >>> end of math interlude <<<

16 The Schrödinger Equation (1926) Outline: A first success: the hydrogen atom Unpacking the Schrödinger equation Infinite well potential Tunnelling

17 The hydrogen atom Bohr-Rutherford 1913 L=n h DeBroglie 1924!"#$%&'()*)+',-&'./0"1"/2'.%/343"5"16'7"01%"3$1"/20'400/8"41&7'9"1-'1-&'04:&'/%3"1450'40'"2'!"#$%&'()*);)'<:4#= "2"2#':&1-/7>'0$%?48&'/?'8/201421'.%/343"5"16)'@/1'1/'0845&) Schrodinger 1926 : 400nm 700nm

18 Which part of don t you understand?

19 Commentary The purpose of the next couple of slides is to investigate the genetic makeup of the Schrödinger equation (SE). The SE occupies a prominent place in physics. It is essentially the quantum counterpart of Newton s axioms. Thus the question arises of whether the SE is completely unrelated to what came before it, or whether concepts identified before in classical mechanics reappear. The latter turns out to be the case! This is background knowledge

20 Unpacking the Schrödinger equation Let s confine to one-dimension settings: i 2 d 2 Ψ(x, t) Ψ(x, t) = t 2m dx 2 + V (x)ψ(x, t). What s the purpose of the derivatives d dx?

21 Commentary: Momentum eigenstates On the next slide, we invoke a momentum eigenstate. That s a quantum state with a sharply defined momentum p. As any quantum state, a momentum eigenstate has a wave function. It looks like this: i 1 x Ψ(x) = e i2πx/λ

22 Unpacking the Schrödinger equation What happens if we apply i d dx to a wave Ψ(x) = ei2πx/λ? i 1 x

23 Unpacking the Schrödinger equation What happens if we apply i d dx to a wave Ψ(x) = ei2πx/λ? We obtain i d 2π Ψ(x) = dx λ Ψ(x). Now using the DeBroglie relation, i d Ψ(x) = p Ψ(x). dx We thus find that applying the operation i dx d to quantum state with sharp momentum keeps the wave function unchanged, with a multiplicative factor p (the momentum!) in front.

24 Unpacking the Schrödinger equation What happens if we apply i d dx to a wave Ψ(x) = ei2πx/λ? i d Ψ(x) = p Ψ(x). dx Based on this observation, we identify i d dx = momentum p This is how the de Broglie relation enters the Schrödinger equation

25 Unpacking the Schrödinger equation What does this term in the Schrödinger equation represent? i 2 d 2 Ψ(x, t) Ψ(x, t) = t 2m dx 2 + V (x)ψ(x, t) A: It represents kinetic energy. i t Ψ(x, t) = [ ˆp 2 2m + V (x) ] } {{ } total energy Ψ(x, t) Commentary: This is the end point of our gene search. We have found something very familiar-looking on the r.h.s. of the SE, the energy function. In complete analogy to classical mechanics, it has a kinetic part and a potential part, and the kinetic part has the exact same momentum dependence as it has in nonrelativistic classical mechanics.

26 Unpacking the Schrödinger equation We thus find that on the right hand side of the SE we have the energy function H, i Ψ(x, t) = HΨ(x, t). t H is called the Hamilton operator, or Hamiltonian. The significance of H is two-fold: H measures the energy of the physical system. H generates the evolution of Ψ in time.

27 The Hamiltonian as the generator of evolution Use the Schrödinger equation with the Euler method: i Ψ(x, t) = HΨ(x, t) t Ψ(x, t + t) = Ψ(x, t) + t HΨ(x, t) i The Hamiltonian H drags the wave function Ψ forward in time.

28 William Rowan Hamilton

29 Commentary: Rowan Hamilton Hamilton died in 1865, and the Schrödinger equation is from How did he smuggle his name in there? It turns out that the energy function generating time evolution is again a concept that has previously appeared in classical mechanics. Hamilton discovered it. There is a way of formulating classical mechanics, so-called Hamiltonian mechanics, which is only an epsilon away from quantum mechanics. So why didn t Hamilton take the final step and invented quantum mechanics? Simply, at the time no piece of observational data called for it. (I confirm Ray here: Hamiltonian mechanics is a 300 subject.)

30 Comment: time-independent Schrödinger equation We are now switching gears. The above was for your background understanding. What s coming up is exam-relevant.

31 Time-independent Schrödinger equation Time-dependent Schrödinger equation: Ansatz: i 2 d 2 Ψ(x, t) Ψ(x, t) = t 2m dx 2 + V (x)ψ(x, t). Ψ(x, t) = e iet/ Ψ(x). An ansatz is a trial construct. It substitutes gambling for honest work, and is one of the physicist s dearest tools. What we do here is to assume that the wave function is of a particular form, and see if it checks out. Now turn to Problem 1 on the worksheet

32 Time-independent Schrödinger equation Time-dependent Schrödinger equation: Ansatz: Ψ(x, t) = e iet/ Ψ(x). i 2 d 2 Ψ(x, t) Ψ(x, t) = t 2m dx 2 + V (x)ψ(x, t). The ansatz did indeed check out! We remain with a constraint on the spatial part of the wave function ψ(x). This constraint is the famous... Time-independent Schrödinger equation: E: total energy. 2 d 2 Ψ(x) 2m dx 2 + V (x)ψ(x) = E Ψ(x)

33 The Ansatz-method instructions for use The Ansatz worked, and all solutions that come from it will be of the assumed form ψ(x, t) = e iet/ ψ(x). However, this does not mean that all solutions of the time-independent SE are of the above form. When making the Ansatz, we ad-hoc inferred a property, and thus reduced the solution space. An easy way to see this is to observe that if ψ(x, t) and ψ (x, t) are solutions of the time independent SE, then so is any linear combination Ψ(x, t) = c ψ(x, t) + c ψ (x, t), with c and c complex numbers [Prove it!]. If both ψ(x, t) and ψ (x, t) are of the Ansatz form, Ψ(x, t) is typically not. [Check: When is it, when not?]

34 Complex numbers, really? Do I need complex numbers to understand quantum mechanics? The pro and con: Without complex numbers... You can understand most of the quantum phenomena of interest for this class: superposition and interference, the Heisenberg uncertainty relation, quantum measurement, bound states, entanglement. You cannot even parse the time-dependent Schrödinger equation, which however is the most fundamental equation in quantum mechanics. Bottom line: Complex numbers are here to stay; better get used to them

35 What s the i good for? If we take the i out of the time-dependent SE, what happens? The SE has the property that, whatever the form of the potential V (x), the norm of the wave function (= integral of absolute value squared) is equal to 1 for all times. If we take the i out, that property is gone. Physical significance of this fact: Let s consider the pointless measurement, asking: Is the particle anywhere in space?. We can predict that the answer is Yes with probability 1, hence no need to measure. If you take the i out of the SE, this is no longer true. The probability to find the particle anywhere in space will either exponentially increase or decrease with time, and be 1 only at one time instant. That doesn t make sense. Hence the i must be in the SE.

36 Let s back up this claim When taking the i out, the time-dependent SE becomes the temporary Sci1-E, which reads 2 d 2 Ψ(x, t) Ψ(x, t) = t 2m dx 2 + V (x)ψ(x, t). As with the SE proper, we make an Ansatz: ψ(x, t) = e Et/ ψ(x). This leaves the proper time-independent SE for ψ(x), as before. Thus, the norm N(t) := ψ(x, t) 2 dx has the time dependence N(t) = N(0)e 2Et/. N(t) is the probability for finding the particle anywhere in space, at time t. At this point, we discard the temporary Sci1-E.