All lecture notes with drawings PDF

Preview

Summary

All lecture notes with drawings., All lecture notes with drawings., All lecture notes with drawings....

Description

Lecture Notes PHYS323 9/20/16 Concluded (for now) discussion of Special Relativity Short mention of General Relativity: Scheme to incorporate gravity into Special Relativity not as a force, but as a curvature of space-time. Now pivoting to Quantum Mechanics (QM). First question: What “obvious” everyday notions about reality do we have to give up on in the light of this 2nd early 20th-century revolution in Physics?

As this screen shot shows, the underlying idea of Classical Mechanics (CM) - including Special Relativity - is that observables like velocity, position, momentum, angular momentum, energy etc. are “real” properties that particles or systems of particles “have” - while we can’t measure them with infinite precision, they do have precise values on a continuum (real numbers). In particular, it makes sense to talk about these quantities as well-defined functions of time - r(t), p(t) etc. Furthermore, Newtonian dynamics connect them among each other - e.g., position determines force, and force in turn determines change of momentum. Similarly for waves, e.g. electromagnetic waves, that can be characterized by continuous, arbitrarily precise (in principle) values of electric and magnetic fields everywhere in space. Quantum Mechanics is more akin to the idea by Plato that we cannot observe the world “as it really is”, but rather we can only derive laws about our observations. Objects like

particles or systems are not characterized by precise values of observables, but rather by their “state vectors” which contain all the information we can have about those objects (from observations). Instead of predicting precise values for observables, these state vectors (a.k.a “wave functions”) only predict probabilities for certain outcomes. Even more strange, some observables can only assume discrete values out of a pre-determined set (like eyes on a die) - not a continuous range of values (example: energy of an electron bound in an atom). Hence, to study QM we must study probability:

Distinguish between discrete outcomes (eyes on a die) with a given probability for each outcome and continuous range of outcomes (wait times for a bus) with a given probability distribution - in that case, the probability of a certain observation with a given range is proportional to the size of that range: ΔProb(x…x+Δx) = p(x) . Δx. In the first case, the probability for a set of outcomes (e.g., “more than 3 eyes” is the sum over the individual outcomes (here: 1/6 + 1/6 + 1/6 = ½), and the sum over all probabilities is 1. In the second case, the probability for an outcome in a larger range is the integral of p(x) over that range, and the integral over the whole possible range of outcomes is 1. Some examples and definitions (see screen shot on next page): - flat distribution (all outcomes equally likely - “Gaussian bell curve” - symmetric around maximum - skewed distribution (long “tail” on one side of maximum For each distribution, one can define the - Mode: The value x for which the distribution (or probability) reaches its maximum; the most likely outcome - Median: The value x for which the probability of a result below x is equally likely (50%) as for a result above x (“split the distribution in the middle”) - Average or Mean or Expectation Value: Best estimate of result (see below)

Can define the expectation value for any function of the observable, f(x): f = ∫ f (x)p(x)dx (the integration goes over the full range of possible outcomes) Expectation value for the variable itself (i.e., its Mean or Average): µ ≡ x = ∫ xp(x) dx Expectation value for the mean square deviation from the mean (known as the variance):

σ2 = (x − µ)

2

=

∫ ( x − µ)

2

p(x)dx = 2

∫x

2

p(x)dx − 2µ ∫ xp(x) dx + µ 2 ∫ p(x)dx =

= x 2 − 2µ x + µ 2 = x 2 − x The square root of the variance is known as the “standard deviation” and provides a measure on the “uncertainty” of the outcome, i.e., how far any given measurement likely will deviate from the average: Δx ≡ σ 2 =

x2 − x

2

Lecture 9/22/2016 The expectation value : is the mean, or average value for a given probability distribution. It is also denoted by the Greek letter µ. For a discrete set of probabilities,

Prob(x (x i ) = ∑ x i Prob

For a continuous set of probabilities, = ∫ x p (x)dx and = ∫ f(x) p (x)dx

We require that

∫p(x)dx=1

“Variance” is denoted by σ 2 = ∫ (x - µ) 2 p (x)dx =

∫ x 2 p (x)dx

- 2µ ∫ x p (x (x)dx )dx + µ 2 ∫ p (x)dx =

- 2µ 2 + µ 2 = - µ = - 2 , with the limits of integration ranging from some Xmin to Xmax. This relationship expresses the difference between and 2 . They are NOT interchangeable. Standard deviation σ = square root of variance Example of Problems where probability is requir ed: Position of and electron: Find the probability that the electron may be found between x=100nm--200nm: Prob(100nm--200nm) ≈ p(150nm)x100 nm. Calculate its expectation value. Example: = 500nm with σ= 100nm. which means 500 nm is the best prediction with an uncertainty of ±100 nm.

Units for Subatomic Physics: 1eV=1.602x10-19J m = E/c2 -> new unit for mass: eV/c2, e.g. melectron= 511,000eV/c2 New unit for Momenta: eV/c h=6.66x10-34J∙s and ћ=h/2π =197.33 nm . eV/c The upper limit to how precise a system’s position and momentum may be specified (predicted or measured) is given as h/4π. This can also be expressed as follows:

≥ћ/2 /2 which is known as the Heisenberg Uncertainty Principle Principle.. σ p σ x ≥ћ Classical Mechanics The defining properties include position (r), momentum (p) and mass (m). p = m(dr/dt)à can predict r(t+Δt) from r(t) and p r -> F = dp/dt -> can predict p(t+Δt) from p(t) and F (r). Quantum Mechanics The defining property is the state vector, |Ψ>. Properties of this “state vector”? 1) Contains ALL information that one CAN have about a particle/system 2) Can be used to predict probability for any measurement outcome 3) Describes how a system evolves in the future: |Ψ>(t) => |Ψ>(t+Δt)

ONE example for a state vector: Functions mapping from the Real Numbers to the complex numbers, Ψ(x). In order to determine a probability for something contained in the state vector, you must multiply by its complex conjugate in order to calculate a non-complex probability. For quantum mechanics to apply the techniques used in probability, we require that

∫Ψ*(x)Ψ(x)dx=1.

Here, the probability density of finding the particle near x is given by p(x) = Ψ*(x)Ψ(x). Because the state vector is complex-valued, it can simultaneously also encode probabilities for other observables (like a flagpole which can have many shadows depending on where the light comes from).

Lectur e 9 /2 7 /1 6 Notes Review: • • •

Quantum Mech. makes predictions about probabilities of states, observables, and measurements. There are two types of measurement outcomes, discrete and continuous. Continuous measurements – Have probability densities, p ( x ) , determines the probability of a measurement result lying in some range: Pr ob(x+ Δ x) ≈ p (x) (x)∆x. x. Every continuous measurement has inherent uncertainty (standard deviation of the predicted outcome). For example, position and momentum have uncertainties σ x and σ p respectively, and are related to each other by the inequality σ x σ p ≥

•

𝒉 𝟒𝝅,

where h is Planch’s constant.

Discrete measurements – Have simple probability, Pr ob(x i ) for a specific outcome x i . Discrete measurements theoretically can be measured to infinite precision.

New material: Examples of discr ete pr probabiliti obabiliti obabilities es es:: 1. Binding energies 𝒉

2. Angular momentum (J J v ect or = n 𝟒𝝅 ; n = 0,1 ,2,… ,2,…) 3. Light From Classical Mech. light is thought of as a continuous wave with wavelength, λ, and frequency, f, which are related by: c = λ f . [ N ot e : we also define a wave “number ” k = 2 2π π / λ and a “angular frequency” ω = 2 π f , with c = ω /k.) In Quantum Mech. light is thought of as made of discrete wave packets called a photons. Einstein theorized correctly in his theory of the photoelectric effect that these packets of light have energy, E=h f and ther efor e mome momentum ntum p = E /c = h/ λ . Photons are generally represented with the symbol γ. h/λ The equation of energy E=h f can be used to explain blackbody radiation. An object with a temperature greater that 0 kelvin has thermal energy. This thermal energy comes from the vibration of atoms. The atomic vibrations fluctuate, and energy is passed from atom to atom or the energy is lost from the object via light. Since the energy that can be emitted via light of given frequency is quantized, Planck could explain why the blackbody spectrum does not become more and more intense at higher and higher frequency - he derived his law for blackbody radiation and also introduced “his” constant, h.

A second example is light emitted from a dilute gas of a specific type of atoms. This light comes from electrons dropping to lower allowed energies within each atom. As stated above, these energies (binding energies) are discrete. Since there can only be specific energy losses, there can only be specific wavelengths of light emitted for a particular object. In objects made of many different atoms the emitted wavelengths of light look as though they are continuous because of the various allowed energy levels in each atom. However, in a pure substance the discreteness of the light wavelengths can be seen.

W ha t is th the e ssta ta tte e v ve e cto ctorr ? The state vector is a function of time that can be used to describe the state of the system and can be used to pr edict the pr obability for me a s u urr in ing g certain values for all obser vables vables. All of the underlined concepts require further explanation.

W ha t is a ve c ctor tor tor? ? A vector is a member of a vector space and can be described as a magnitude and direction or as a list of numbers. The list of numbers must be: • •

Addable:: 𝒓𝟏 + 𝒓𝟐 = (x 1 + x 2 , y 1 + y 2 , z 1 + z 2 ) Multipliable with a scalar: 𝒂𝒓𝟏 = (ax 1 , ay 1 , az 1 )

A vector space has dimensions. The number of dimensions a vector space can have is any positive number n:: n = 1 ,2 ,3 ,,… … , ‫ 𝟎א‬,…, ‫א‬ ‫ = 𝟎א‬coun countable table ∞ ‫ = א‬conti continuous nuous ∞ A f uncti unction on ff(x) (x) is a continuous vector where x ∈ ℝ ( real numbers) and f (x) ∈ ℂ ( complex vector space). All functions f(x) form a vector space. Special Vector Spaces: “Scalar Product” A special vector space is one where the dot product can be defined. EX:

cos( cos(α α) 𝒓𝟏 ∙ 𝒓𝟐 = |𝒓𝟏 ||𝒓𝟐 |cos(

In a discrete vector space: {a i , i = 1 ,2,3,…, ∞}} ∙{{ b i , i = 1,2,3,… , ∞}} = In a continuous vector space: f ∙g =

! 𝒊!𝟏𝒂𝒊 𝒃𝒊

𝒇∗ (𝐱)𝒈(𝐱) 𝒅𝒙

A vector space w ith compl comple e x s ca llar ar s and a w ell ell-- defined scal scalar ar pr oduct iis s called a Hilbert space. All state vectors are members of a suitable Hilbert Space.

Example Example: Describe position along the x-axis. Hilbert Space is made of vectors |ψ> defined by complex-valued functions ψ(x) that are “square-integrable” (to ensure the existence of the scalar product: =

𝝓∗ (𝐱)𝝍(𝐱) 𝒅𝒙 < ∞

In particular, the probability density for finding the particle described by |ψ> near some position x is given by p(x) = 𝝍∗ (𝐱)𝝍(𝐱).. But the same state vector also encodes information about all other observables one could choose to measure, e.g. momentum.

PHYSICS 323 – Fall Semester 2016 - ODU September 29, 2016 Modern Physics –Lecture 10 Notes A state vector contains all knowable information about a state:

ℂ ∋ ψ 𝑥 , 𝑥∈ ℝ

Vectors in a vector space ψ t can be added → ψ! + ψ!

ψ x +φ x

superposition

can be multiplied with z∈ ℂ: ψ! → z ψ!

(same state)

has a scalar product : ψ! , ψ! → 〈ψ! |ψ! 〉 Example: Function ψ x represents state vector for particle moving along x-axis. If state vector changes as time passes, we can write it as a function ψ x, t . To compare to a prototypical wave: 𝑓 𝑥, 𝑡 = 𝑒 !(!"!!") , 𝑣!!!"# = Prob x … x∆x =

! !

!

First calculate ψ

, then define |ψ!"# 〉=

! !

Example: Schr’s cat *) ψ Ex.) ψ! x

*)

!! !

, 𝜔 = 2𝜋𝑣)

i. ψ∗ x ψ x dx ii. [ψ∗ x + φ∗ (x)] [ψ x φ(x)] = ψ∗ x ψ x + φ∗ x φ x + ψ∗ x φ x + φ∗ x ψ x = | ψ x ! + φ x ! + 2𝑅𝑒 φ∗ x ψ x INTERFERENCE!

𝑐 ∙ ψ describes the same state as ψ and so can be normalized ψ

For probabilities

, (𝑘 =

ψ!

!

=

ψ!"# 𝑥 =

!

! !!

= c!∗ , c!∗

!! !!

!

!

= 1.

|ψ〉.

= |c! |! + |c! |! , |ψ!"# 〉 =

! | !! | ! ! | !! | !

ψ∗! x ψ! x 𝑑𝑥 < ∞ !

! ∗ !! !!

! !! ! !"

∙ ψ! x

Ex. 2-D Hilbert space:) S.s.’s cat: most general, 𝑐! ↑ +𝑐! ↓, 𝑐! , 𝑐! = ψ , P ↑ = |c! |! , P ↓ = c! ! !

initial state: ↑ 𝑎𝑙𝑖𝑣𝑒

𝑑

𝑎 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑖𝑜𝑛 𝑆𝑐ℎ𝑟𝑜𝑑𝑖𝑛𝑔𝑒𝑟 𝐸𝑞𝑛: 𝑖ħ 𝑑𝑡 ψ = Hψ

! !

↑+

! !

(𝑡ℎ𝑒 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦: 𝑝𝑟𝑜𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑛 𝑒𝑖𝑔𝑒𝑛𝑠𝑡𝑎𝑡𝑒 (𝑎𝑛 𝑒𝑞𝑢𝑎𝑙𝑙𝑦 𝑑𝑒𝑎𝑑 𝑎𝑛𝑑 𝑎𝑙𝑖𝑣𝑒 𝑐𝑎𝑡)) Observables can be chosen to be measured. A measurement instantaneously changes the state vector (either ↑ or ↓), causing a ‘collapse’ of the wave function into an eigenstate of the operator.

↓

Physics Notes 10-25-16 Separation of variables 1D

3D

Ψ(x,t)

Ψ(x,y,z,t)

Prob: (x….x+ Δx) | Ψ(x,t)| 2 * Δx

Prob. (x…. x+ Δx, y…..y+Δy, z….z+Δz) (inside a 3-D volume Δτ=ΔxΔyΔz) | Ψ(x,y,z,t)|2 * Δx* Δy*Δz

Require: ∫ |Ψ(x,t)|2 dx=1

∫∫∫|Ψ(x,y,z,t)|2 dxdydz=1

3 position operators X,Y,Z (X Ψ )(x,y,z) = x * ϕ(x,y,z) (Y Ψ )(x,y,z) = y * ϕ(x,y,z) (Z Ψ )(x,y,z) = z * ϕ(x,y,z) Momentum P

vector form P= (h/i)

Px (h/i) d/dx Py (h/i) d/dy Pz (h/i) d/dz H=(Px2 + Py2 + Pz2 )/(2m) + V(x,y,z) = (-h2/2m)(d2/dx2 V(x,y,z)* Ψ(x,y,z) =H Ψ To Do: 1. Find eigenstates H ϕE (x,y,z) = E ϕE(x,y,z) 2. Write full solution to Schrödinger equation

Ψ

+ d2/dy2

Ψ

+ d2/dz2 Ψ) +

V=(x,y,z) else

0 if 0 < x < L and 0 < y < L and 0 < z < L and infinity

Separation of Variables: ϕE(x,y,z)= ϕ1(x) ϕ2(y) ϕ3(z) [-h2/2m][(d2 ϕ1/dx2) ϕ2 ϕ3 + (d2 ϕ2 /dy2) ϕ1 ϕ3 + (d2ϕ3/dz2) ϕ1 ϕ2) + V*ϕ1 ϕ2 ϕ3= E* ϕ1 ϕ2 ϕ3 [-h2/2m][(1/ϕ1(x))(d2ϕ1/dx2) + (1/ϕ2)(d2ϕ2/dy2) + (1/ϕ3)(d2ϕ3/dz2)] + V = E F(x)=E1

G(y)=E2

H(z)=E3

3 Equations (-h2/2m)(d2 Ψ(x)/dx2)=Eϕ1(x) L, 0 else

ϕ1(x)= Asin(n πx/L), E1=(n2 π2 h2/2mL2) 0 < x <

(-h2/2m)(d2 Ψ(y)/dy2)=Eϕ2 (y)

ϕ2(x)= Asin(m πy/L),E2=(m2 π2 h2/2mL2)

(-h2/2m)(d2 Ψ(z)/dz2)=Eϕ3 (z)

ϕ3(x)= Asin(k πz/L), E3=(k2 π2 h2/2mL2)

Solution: ϕn,m,k(x,y,z) = Asin(n πx/L) sin(m πy/L)sin(k πz/L) Quantum Numbers n,m,k=1,2,3…. En,m,k = (π2 h2/2mL2)(n2+m2+k2) Lowest (ground state) energy E1,1,1 = 3(π2 h2/2mL2) Degeneracy: Same energy for several different states, e.g. E1,1,2 = E1,2,1 = E2,1,1

Physics 323 October 27, 2016 Q ua ntum Me cha chanics nics iin n 3 D – A cha nge to pola r coor di dina na te s We can write Schrodinger’s equation in the stationary form if we solve the following equation. 𝑑! 𝑑! ℏ! ! 𝑑! − + !+ ! ∇ 𝜑! 𝑟 + 𝑉 𝑟 𝜑! 𝑟 = 𝐸𝜑! 𝑟 𝑤ℎ𝑒𝑟𝑒 ∇! = ! 2𝑚 𝑑𝑧 𝑑𝑦 𝑑𝑥 The full, time-dependent version then has the solution !

!

𝜑! 𝑟, 𝑡 = 𝜑! 𝑟 𝑒 ℏ!" However, many times in nature the potential does not depend individually on x, y, and z. Instead, it depends only on the distance from some fixed point. That is, it’s spherically symmetric. One such example is electric potential energy, given by 𝑄𝑞 1 𝑉 𝑟! = 4𝜋𝜀! 𝑟 The more general form of the potential is 𝑉 𝑟 ; 𝑟 = 𝑥 ! + 𝑦 ! + 𝑧 ! The problem that arises here is that this gives us an equation in terms of a combination of x, y, and z. A more appropriate approach would be to convert these equations to spherical coordinates, r, θ, ϕ. Z ( x,y ,z A line from an origin to a point in three dimensional space forms a Y vector, r. The projection of this vector onto the x,y plane has an angle 𝜑 with respect to the x axis. Likewise, the has an angle 𝜃 with the θ z axis. With a little trigonometric intuition, it is easy enough to see in ฀ the picture that the coordinates X (x,y,z) of the point can individually be represented with sine and cosine, where: 𝑥 = 𝑟𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜑 𝑦 = 𝑟𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜑 𝑧 = 𝑟𝑐𝑜𝑠𝜃 Intuitively, this tells us to look for a solution where the eigenstate is represented in terms of r, θ, ϕ. The only thing left to do is to represent the gradient squared operator in terms of our newfound equations for x,y, and z. ∇! =

𝑑𝑥 !

𝑑!

𝑑!

𝑑! +

𝑑𝑦 !

+

𝑑𝑧 !

=

1 1 1 𝜕 1 𝜕 ! 𝜕 𝜕! 𝜕 𝑠𝑖𝑛𝜃 + + 𝑟 𝜕𝜃 𝑟 ! 𝑠𝑖𝑛! 𝜃 𝜕𝜑 ! 𝑟 ! 𝜕𝑟 𝜕𝑟 𝑟 ! 𝑠𝑖𝑛𝜃 𝜕𝜃

Substituting this back into the original equation and expanding the coefficient of the gradient squared operator leaves us with:

−

ℏ! ℏ! ℏ! 1 𝜕 1 𝜕 1 𝜕! 𝜕 𝜕 𝜑! 𝑟 , 𝜃 , 𝜑 + 𝑉 𝑟 𝜑! 𝑟 , 𝜃, 𝜑 = 𝐸𝜑! 𝑟 , 𝜃 , 𝜑 𝑠𝑖𝑛𝜃 − − 𝑟! 2𝑚 𝑟 ! 𝜕𝑟 𝜕𝑟 2𝑚𝑟 ! 𝑠𝑖𝑛𝜃 𝜕𝜃 𝜕𝜃 2𝑚𝑟 ! 𝑠𝑖𝑛! 𝜃 𝜕𝜑!

What a mess! We need to find a way to clean this up a bit. The RHS is a constant and the left hand side is split into two parts: one part depending on r and its derivatives and one part depending on theta and its derivatives. This would lead us to look for a function for the eigenstate to be expressed as a product of two separate functions: one function is expressed solely in terms of r and the second function is expressed in terms of theta and phi: 𝜑! (𝑟, 𝜃, 𝜑) = 𝑅 𝑟 𝑌(𝜃, 𝜑) We can perform a separation of variables on the equation and introduce these two new functions, and divide through by ϕE. 1 𝑅 𝑟

−

1 ℏ! 1 𝜕 ! 𝜕 𝑅 𝑟 + 𝑟 𝑌 𝜃, 𝜑 2𝑚 𝑟 ! 𝜕𝑟 𝜕𝑟

−

1 𝜕 ℏ! 𝜕 ℏ! 𝜕 ! 𝑠𝑖𝑛𝜃 + 𝑌 𝜃, 𝜑 ! 2𝑚𝑟 𝑠𝑖𝑛𝜃 𝜕𝜃 𝜕𝜃 𝑠𝑖𝑛! 𝜃 𝜕𝜑!

= 𝐸 − 𝑉(𝑟)

Since the first term on the lhs and the rhs are both functions of r only, the part of the lhs dependent on theta and phi (inside the square brackets) must be actually independent of theta and phi and is therefore a constant. This leads us to find a solution such that, 𝑌 𝜃, 𝜑 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑌(𝜃, 𝜑) To solve this equation would require weeks of higher level math, so we take a simple case where θ = 90°. This removes theta from the equation because the sine of 90° is 1, and since theta is constant its derivative is zero. We’re left with: −ℏ!

𝜕! 𝐹 𝜑 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝐹(𝜑) 𝜕𝜑!

So we’re looking for a function whose second derivative multiplied by -ℏ2 yields a constant multiplied by that same function. A possible solution is the natural exponential function,

𝐹 𝜑 = 𝑒 !"# This means that the constant on the RHS is (-α2)(-ℏ2). However, we’re not done. This equation introduces to us yet another problem. The variable phi goes from 0 to 2π, which are the same point (meaning ϕ + 2π = ϕ). We must require that

𝑒 !"(!!!!) = 𝑒 !"# For this to be true, α has to be some integer, m, where m = 0, ±1, ±2, ±…, ±∞. This requirement tells us that the solution we found is quantized. It can only exist in multiples of ℏ, similar to the square well. The next part requires a gigantic leap of faith as it’s the very, very heavy math intensive portion. It all leads to finding the angular momentum operator(s).

Angula r Mome ntum O pe rra a tor (s) If we look at the differential operator above, it turns out to be the angular momentum operator compone...