ħ ≡ h/(2π)
ħ: Planck's constant
h: Planck's original constant
.
Schrodinger equation :
, Ψ := Ψ(x,t)
.
Question: Suppose I do measure the position of the particle, and I find it to be at point C. Where was the particle just before I made the measurement?
Answers:
- 1. (Einstein)The realist position: The particle was at C. . To the realist,
indeterminacy is not a fact of nature, but a reflection of our ignorance.
"the position of the particle was never indeterminate, but was merely unknown
to the experimenter." Evidently Ψ is not the whole story—some additional
information (known as a hidden) is needed to provide a complete description
of the particle.
- 2. (Copenhagen)The orthodox position: The particle wasn't really anywhere. It was the act
of measurement that forced the particle to "take a stand" (though how and why it
decided on the point C we dare not ask). Jordan said it most starkly: "Observations
not only disturb what is to be measured, they produce it ... We compel (the
particle) to assume a definite position." Among physicists it
has always been the most widely accepted position. Note, however, that if it is
correct there is something very peculiar about the act of measurement—something
that over half a century of debate has done precious little to illuminate. ()
- 3. The agnostic position: Refuse to answer. (Denied by Bell's Inequality)
In 1964 John Bell astonished the physics community by showing
that it makes an observable difference whether the particle had a precise (though
unknown) position prior to the measurement, or not.
Bell's argument; A particle simply does not have a precise position prior to measurement, any
more than the ripples on a pond do; it is the measurement process that insists on
one particular number, and thereby in a sense creates the specific result, limited
only by the statistical weighting imposed by the wave function.
There are, then, two entirely distinct kinds of physical processes:
- "ordinary" ones, in which the wave function evolves in a leisurely fashion under the Schrodinger equation,
- "measurements," in which Ψ suddenly and discontinuously collapses.
S equation 中的wave function Ψ之所以可以解释为概率波的原因：
- ∫{+∞,-∞}|Ψ(x,t)|^2 dx = 1, |Ψ(x,t)|^2视为概率密度
- AΨ满足S equation，
- normalized the wave function will stay normalized in S equation:
d/dt (∫{+∞,-∞}|Ψ(x,t)|^2 dx) = 0
.
|Ψ|^2 = Ψ* Ψ
==>
J is called the probability current(概率对时间的变化率)
dP_ab/dt = J(a,t)-J(b,t)
all classical dynamical variables can be expressed in terms of position and momentum.
.
de Broglie formula: p = h/λ =2πħ/λ
.
uncertainty principle: σx σp >=ħ/2,
(σx=sqrt(<x^2>-<x>^2), σp=sqrt(<p^2>-<p>^2))
-------------------------------------------------
CH2 Ψ(x,t)=ψ(x)φ(t)
.
iħ 1/φ dφ/dt := E ==>φ(t)=exp(-iEt/ħ)
==>
ψ(x)=...
==> Ψ(x,t)=ψ(x) exp(-iEt/ħ)
(E: total energy)
.
What's so great about separable solutions(Ψ(x,t)=ψ(x)φ(t))?
1. They are stationary states.
2. They are states of definite total energy. every measurement of the total energy gets the same value. 能量守恒
3. The general solution is a linear combination of separable solutions.
Once you've solved the time-independent Schrodinger equation, you're essentially done; getting from there to the general solution of the time-dependent Schrodinger equation is, in
principle, simple and straightforward.
|c_n|^2: the probability that a measurement of the energy would yield the value En. So ∑|c_n|^2 = 1
In classical mechanics, the total energy (kinetic plus potential) is called the Hamiltonian:
H(x,p) = p^2/(2m) +V(x)
Hamiltonian operator:
<H>= ∑ |c_n|^2 En
The Infinite square well
,
ψ=A*sin(kx)
==> kn=nπ/a,
En= 1/(2m)*(nh/a)^2
.
2.3 THE HARMONIC OSCILLATOR (V(x)=1/2 m(ωx)^2)
But practically any potential is approximately parabolic, in the neighborhood of a local minimum. That's why the simple harmonic oscillator is so important: Virtually any oscillatory motion is approximately simple harmonic, as long as the amplitude is small.
.
ladder operators:
raising operator: a_+=(-ip+mωx)/sqrt(2ħmω)
lowering operator: a_-=(+ip+mωx)/sqrt(2ħmω) (a_+的共轭)
.
x and p operator:
x = sqrt(ħ/2mω)(a_+ + a_-)
p = i sqrt(ħmω/2)(a_+ - a_-)
.
a_-a_+
= 1/(2ħmω)*[p^2+(mωx)^2] –i/(2ħ)[x,p]
= 1/(2ħmω)*[p^2+(mωx)^2] +1/2
≡ H/(ħω) + 1/2
==> H = ħω(a_-a_+ -1/2) = ħω(a_+a_- +1/2)
.
commutator of A and B: [A,B]≡AB-BA
.
[x,p] = iħ (canonical commutation relation)
[a_-,a_+] = 1, [a_+,a_-] = -1,
.
the Schrodinger equation19 for the harmonic oscillator: Hψ=Eψ
If Hψ=Eψ, then H(a_+ψ) = (E+ħω)(a_+ψ) and H(a_-ψ) = (E-ħω)(a_-ψ)
So, if you get a solution ψ, use (cn*a_+ψ) and (dn*a_-ψ) to generate more solutions.
.
a_- ψ_0 = 0 (P46, 但没看懂)
==> d ψ_0/dx = -mω/ħ x ψ_0
==> ψ_0 = A exp(-mω/(2ħ) x^2) （normalize，求得A）
同时，由a_- ψ_0 = 0 and Hψ_0=E_0ψ_0 ==> E_0 = ħω/2
所以， ψ_n(x)=A_n (a_+)^n ψ_0(x), E_n=(n+1/2) ħω
A_n=1/sqrt(n!)是normalize ψ_n 得到的。
。
.
a_+ψ_n = sqrt(n+1)ψ_{n+1}
a_- ψ_n = sqrt(n)ψ_{n-1}
.
ψ_n = 1/sqrt(n!) (a_+)^n ψ_0
.
.
2.4 THE FREE PARTICLE (V(x)=0)
k≡sqrt(2mE)/ħ
separable solutions： Ψ(x,t) = A exp[i(kx -hk^2t/2m)]
但这个波函数无法单位化，对此的解释：
the separable solutions do not represent physically realizable states. A free particle cannot exist in a stationary state; or, to put it another way, there is no such thing as a free particle with a definite energy.
。
但general
solution是可以单位化的：
给定
,
由
Plancherel's
theorem 求得φ(k)：
Plancherel's theorem：
/。
In quantum mechanics, when V = 0, the exponentials represent traveling waves, and are most convenient in discussing the free particle, whereas sines and cosines correspond to standing waves, which arise naturally in the case of the infinite square well.
。
Kronecker delta: δ_ij discrete
Dirac delta: δ(x) continuous
<f_m | f_n> = δ_mn
<f_p| f_p'> = δ(p-p') (Dirac orthonormality)
-----------------------------------
CH 3
Quantum theory is based on two constructs: wave functions and operators.
- The 【state】 of a system is represented by its 【wave function】, wave functions satisfy the defining conditions for abstract vectors,
- Observables are represented by operators. operators act on them as linear transformations.
.
But the "vectors" we encounter in quantum mechanics are (for the most part)
functions, and they live in infinite -dimensional spaces.
.
The collection of all functions of .v constitutes a vector space,
。
<f|g>≡∫f*(x)g(x)
<f|g> = <g|f>*
<af|bg> = a*b<f|g>
。
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
由 ∫|Ψ|^2 = ∫Ψ*Ψ，推广得∫f*(x)g(x)
又有<a|b>= a1*b1 + a2*b2 +a3*b3, 为了和这个类比，定义<f|g>≡∫f*(x)g(x),每个点x是一个维度。
<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
.
Hilbert space: L2(a, b): The set of all square-integrable functions on interval [a, b]
L2(a, b) = {f(x)| ∫{a,b} |f(x)|^2 dx < ∞ (收敛) }
.
in Hilbert space two functions that have the same square integral are considered equivalent. Technically, vectors in Hilbert space represent equivalence classes of functions.
.
The expectation value of
an observable Q(x,p):
.
hermitian operator (一类算子)
Q: the operator representing obsetvables have the very special
property that
(a hermitian operator can be applied either to the first member of an inner product or to the second,)
.
hermitian operators naturally arise in quantum mechanics because their expectation
values are real: Observables are represented by hermitian operators.
。
Q†: the hermitian conjugate of an operator Q, <f|Qg>=<Q†f|g>
.
Determinate states are eigenfunctions of Q.
Measurement of Q on such a state is certain to yield the eigenvalue, q.
定义Zero does not count as an eigenfunction
The collection of all the eigenvalues of an operator is called operator‘s 【spectrum】.
.
3.3 EIGENFUNCTIONS OF A HERMITIAN OPERATOR
- If the spectrum is discrete(e.g. the Hamiltonian for the harmonic oscillator). eigenfunctions lie in Hilbert space and they constitute physically realizable states.
- If the spectrum is continuous(e.g. the free particle Hamiltonian). the eigenfunctions are not normalizable, and they do not represent possible wave functions (though linear combinations of them—involving necessarily a spread in eigenvalues—may be normalizable).
- If the spectrum has both a discrete part and a continuous part(e.g. the Hamiltonian for a finite square well )
3.3.1 If hermitian operator has Discrete Spectra
Theorem 1: Their eigenvalues are real.
Theorem 2: Eigenfunctions belonging to distinct eigenvalues are orthogonal.
.
In finite-dimensional vector space the eigenvectors of a hermitian matrix have a third fundamental property: They span the space.
.
Axiom: The eigenfunctions of an observable operator are complete: Any function (in Hilbert space) can be expressed as a linear combination of them.
.
3.3.2 If hermitian operator has Continuous Spectra
the eigenfunctions are not normalizable(L2(a,b)不收敛), they are not in Hilbert space and
they do not represent possible physical states; nevertheless, the eigenfunctions with
real eigenvalues are Dirac orthonormalizable and complete
3.4 GENERALIZED STATISTICAL INTERPRETATION
If you measure an observable Q(x, p) on a particle in the state Ψ(x, t), you are certain to get one of the eigenvalues of the hermitian operator Q^(x. p^).
If the spectrum of Q is discrete, the probability that a measurement of Q would yield the value q_n (associated with the orthonormalized eigenfunction f_n(x)) is |c_n|^2 where c_n=<f_n|Ψ>.
Upon measurement, the wave function "collapses" to the corresponding eigen-state.
If the spectrum of Q is continuous, with real eigenvalues q(z) and associated Dirac-
orthonormalized eigenfunctions f_z(x), the probability that a measurement of Q would yield the value q(z) in the range dz is |c(z)|^2 dz where c(z)=<f_z|Ψ>.
Upon measurement, the wave function "collapses" to a narrow range about the measured value, depending on the precision of the measuring device.
对|c(z)|^2的解释：
|c(z)|^2 is the probabilily that a particle which is 【now】 in the stale Ψ 【will】be in the stale f_n subsequent to a measurement of Q
。
the 【momentum space wave function】 Φ(p, t)，
the 【position space wave function】 Ψ(x,f) (即通常我们所说的波函数)，两者关系：
3.5 THE UNCERTAINTY PRINCIPLE(翻译为“测不准原理”更合适一些)
the statistical interpretation of quantum theory(P106, Sect3.4) ==> uncertainty principle。证明过程P110，sect 3.5
.
For any observable A, B:
There is, in fact, an "uncertainty principle" for every pair of observables whose operators do not commute—we call them 【incompatible observables】. Incompatible
observables do not have shared eigenfunctions—at least, they cannot have a
complete set of common eigenfunctions (see Problem 3.15). By contrast, compatible
(commuting) observables do admit complete sets of simultaneous eigenfunctions（根本原因：This corresponds to the fact that noncommuting matrices cannot be simultaneously diagonalized (that is, they cannot both be brought to diagonal form by the same similarity transformation), whereas commuting hermitian matrices can be simultaneously diagonalized.）
.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
Q:如何判断两个 observable A, B是 incompatible还是 compatible？
A and B is compatible if [A^B^]=0,
A and B is incompatible if [A^B^]!=0,
类比：
- vector |a>,|b > are independent;
α|a>+β|b> spans the whole vector space;
|a>|b>(内积)衡量的是两者的相关程度，|a>|b>=0表示不相关；
|a>,|b> 共有的基向量{ei}不可能span the whole vector space；即总有一个向量v无法被{ei}表示。
- observable x, p are incompatible;
Q(x, p) can express any the variables in class mechanics;
[x^p^] 衡量的是相互干涉的程度, [x^p^]=0表示相互无干涉(无干涉才可兼容，即compatible)；
x,p 共有的那些特征函数{fn}不可能构成一个complete set；即总有一个状态Ψ无法被{fn}表示。
<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
。
对测不准原理的阐述：
- You can certainly measure the position
of the particle, but the act of measurement collapses the wave function to a narrow
spike, which necessarily carries a broad range of wavelengths (hence momenta)
in its Fourier decomposition.
- If you now measure the momentum, the state will
collapse to a long sinusoidal wave, with (now) a well-defined wavelength—but
the particle no longer has the position you got in the first measurement.
- The problem, then, is that the second measurement renders the outcome of the first
measurement obsolete. (第二个观测量干扰(干涉)第一个观测量；或者说第二次观测干扰了第一次的观测结果)
- Only if the wave function were simultaneously an eigenstate
of both observables would it be possible to make the second measurement without
disturbing the state of the particle (即两个观测量相互无干扰(无干涉))(the second collapse wouldn't change anything, in that case).But this is only possible, in general, if the two observables are compatible.
。
测不准原理的很多实验例子：Bohr's article in Albert Einstein: Philosopher-Scientist, edited by P. A. Schilpp, Tudor, New York (1949).
。
some incompatible observables：
- x, p
- x, H=p^2/(2m)+V
- t, E
|a>=(a1, a2, ..., an)^T <a| = (a1*, a2*, ..., an*)
if |a> is a normalized vector, P^≡|a><a|
----------------------------
CH 4 QUANTUM MECHANICS IN THREE DIMENSIONS
.
canonical commutation relations:
[r_i, p_j] = - [p_i, r_j] = iħ δ_ij
[r_i, r_j] = 0 (r1, r2, r3之间无干涉)
[p_i, p_j] = 0 (p1, p2, p3之间无干涉)
.
.
Heisenberg's uncertainty principle in three dimensions:
σ_x σ_px >= ħ/2
σ_y σ_py >= ħ/2
σ_z σ_pz >= ħ/2
abbv. σ_ri σ_pi >= ħ/2
but there is no restriction on σ_ri σ_pj
>>>>>>>>>>>>>>>>>>>>>
可以这么理解： 虽然σ_ri σ_pi 相互干涉，但σ_pi σ_pj无干涉，所以之间无干涉。
<<<<<<<<<<<<<<<<<<<<<
hydrogen stationary states wave functions Ψ(n, l, m).
n≡j_max +l+1. n is called principal quantum number,
l: azimuthal quantum number
m: the magnetic quantum number
n determines the state energy. l and m relates to orbital angular momentum.
|L> = |r> x |p>
fundamental commutation relations for angular momentum：
[Lx,Ly] = iħLz; [Ly,Lz] = iħLx; [Lz,Lx] = iħLy; (cyclic permutation)
two kinds of angular momentum:
- orbital |L>=|r>⨯|p>, the motion of the center of mass,（公转）
-spin S=Iω, the motion about the center of mass. （自转/自旋）
The electron (as far as we know) is a structureless point particle, and its spin angular
momentum cannot be decomposed into orbital angular momenta of constituent
parts (see Problem 4.25).25 Suffice it to say that elementary particles carry intrinsic
angular momentum (S) in addition to their ''extrinsic" angular momentum (L),
。
[Sx, Sy] = iħSz; [Sy, Sz] = iħSx; [Sz, Sx] = iħSy; (cyclic permutation)
S^2 |sm> = ħ^2 s(s+1)|sm>
S_z |sm> = ħ m |sm>
s：spin number
m：quantum number
.
S_± ≡ S_x ± iS_y
S_± |sm> = ħ sqrt(s(s+1)-m(m±1)) |s(m±1)>
（ where s=0, 1/2, 1, 3/2,...; m=-s,-s+1,...,s-1,s）
.
pi mesons have spin 0;
electrons/proton have spin 1/2;
photons have spin 1;
deltas have spin 3/2;
gravitons have spin 2
.
4.4.1 Spin ½
two eigenstates:
- spin up |½, ½>: ↑, χ+=(1 0)^T, λ+= ħ/2
- spin down |½, - ½>: ↓, χ- =(0 1)^T, λ-= -ħ/2
.
the general state of a spin-1/2 particle: χ= aχ+ + bχ-
.
spin operator S
由S^2 |sm> = ħ^2 s(s+1)|sm>, S_z |sm> = ħ m |sm>得：
由S_± |sm> = ħ sqrt(s(s+1)-m(m±1)) |s(m±1)>得：
p176, spin½+uncertainty principal
.
A spinning charged particle constitutes a magnetic dipole. Its magnetic dipole
moment μ=γS
γ: gyromagnetic ratio
.
torque τ=μ⨯B,
The energy associated with this torque is H=- μ∙B= -γB∙S
so the Hamlltonian of a spinning charged particle at rest is Htoo.
。
If you combine spin s1with spin s2, what total spins s can you get?
s=(s1+s2), (s1+s2-1), (s1+s2-2), …., |s1-s2|
----------------------------
CH 5
两个粒子的系统
Quantum mechanics neatly accommodates the existence of particles that are
【indistinguishable in principle】: We simply construct a wave function that is non-
committed as to which particle is in which state. There are actually two ways to
do it:
ψ_+(r1, r2) = A[ψ_a(r1)ψ_b(r2)+ψ_b(r1)ψ_a(r2)] (bosons: integer spin)
ψ_- (r1, r2) = A[ψ_a(r1)ψ_b(r2)- ψ_b(r1)ψ_a(r2)] (fermions: half-integer spin)
This connection between spin and statistics (as we shall see, bosons and fermions
have quite different statistical properties) can be proved in relativistic quantum
mechanics; in the non-relativistic theory it is taken as an axiom.
.
If ψ_a= ψ_b, then ψ_- (r1, r2)=0. ==>Pauli exclusion principal
.
define exchange operator P: Pf(r1,r2) = f(r2, r1)
。
symmetrization requirement：
For bosons, the wave function is required to satisfy : ψ(r1, r2) = ψ(r2, r1) (symmetric)
For fermions, the wave function is required to satisfy: ψ(r1, r2)=-ψ(r2, r1) (antisymmetric)
.
----------------------------------
10.2.3 The Aharonov-Bohm Effect(the vector potential can affect the quantum behavior of a charged particle, even when it is moving through a region in which the |E> and |B> are zero)
In classical electrodynamics the potentials (φ and A)are not directly
measurable—the physical quantities are |E> and |B>:
The fundamental laws (Maxwell's equations and the Lorentz force rule) make no
reference to potentials, which are (from a logical point of view) no more than
convenient but dispensable theoretical constructs. Indeed, you can with impunrcy
change the potentials(gauge transformation):
In quantum mechanics the potentials play a more significant role, for the
Hamiltonian is expressed in terms of φ and A:
quantum theory is still invariant under gauge transformations
-------------------------------------
CHAPTER 12 AFTERWORD
If the electron is found to have spin up, the positron must have spin down, and
vice versa. Quantum mechanics can't tell you which combination you'll get, in
any particular pion decay, but it does say that the measurements will be
correlated, and you'll get each combination half the time (on average).
。
To the realist, there's nothing surprising in this—the electron really had
spin up (and the positron spin down) from the moment they were created ...
it's just that quantum mechanics didn't know about it.
。
You might be tempted to propose that the collapse of the wave
function is not instantaneous, but "travels" at some finite velocity. However, this
would lead to violations of angular momentum conservation,
。
Problem 12.1 entangled state: a two-particle state that cannot be expressed as the product of two one-particle states
。
Bell Therom(1964):
any local hidden variable theory is incompatible with quantum mechanics.
具体来说：For arbitrary orientations, quantum mechanics predicts: P(a,b) = -a∙b. This prediction is incompatible with Bell's inequality(P425).
。
Bell inequality: |P(a,b) – P(a,c)| <= 1+P(b,c)
。
With Bell's modification, then, the EPR paradox proves something far more
radical than its authors imagined: If they are right, then not only is quantum
mechanics incomplete, it is downright wrong. On the other hand, if quantum mechanics
is right, then no hidden variable theory is going to rescue us from the nonlocality
Einstein considered so preposterous. Moreover, we are provided with a very simple
experiment to settle the issue once and for all.
。
Suppose further that the outcome of the electron measurement is independent of
the orientation (b) of the positron detector—which may, after all, be chosen by the
experimenter at the positron end just before the electron measurement is made, and
hence far too late for any subluminal message to get back to the electron detector.
(This is the locality assumption.)这就是《量子力学揭秘》里用扑克牌解释的道理。
。
在实验中：To exclude the remote possibility that the positron detector might somehow "sense" the orientation of the electron detector, both orientations were set quasi-randomly
after the photons were already in flight.
。
震惊科学界的原因：But not because it spelled the demise
of "realism''—most physicists had long since adjusted to this (and for those who
could not, there remained the possibility of nonlocal hidden variable theories.
to which Bell's theorem does not apply). The real shock was the
demonstration that nature itself is fundamentally nonlocal. Nonlocality, in the form of the
instantaneous collapse of the wave function (and for that matter also in the sym-
metrization requirement for identical particles) had always been a feature of the
orthodox interpretation, but before Aspect's experiment it was possible to hope
that quantum nonlocality was somehow a nonphysical artifact of the formalism,
with no detectable consequences. That hope can no longer be sustained, and we
are obliged to reexamine our objection to instantaneous action-at-a-distance.
Influence v.s. cause
Does the measurement of the electron influence the outcome of the positron measurement? ---Yes
does the measurement of the electron cause a particular outcome for the positron?---No
.
two types of influence:
- Causal influence which produce actual changes in some physical property of the receiver, detectable by measurements on that subsystem alone.
Causal influences cannot propagate faster than light,
- Ethereal influence which do not transmit energy or information, and for which the only evidence is a correlation in the data taken on the two separate subsystems—a correlation which by its nature cannot be detected by examining either list alone.
There is no compelling reason why ethereal influences should not propagate faster than light,
.
The influences associated with the collapse of the wave function are ethereal , and the fact that they "travel" faster than light may be surprising, but it is not, after all, catastrophic.
.
12.3 THE NO-CLONE THEOREM
state 的线性叠加
未来的问题：
the nature of measurement and the mechanism of collapse.
-------------------------
¹²³⁴⁵⁶⁷⁸⁹ ᵃᵇᵈᵉᶠᵍʰⁱʲᵏˡᵐⁿᵒᵖ ʳˢᵗᵘᵛʷˣʸᶻ ᵅᵝᵞᵟᵠ ₀₁₂₃₄₅₆₇₈₉ ₐ ₑ ₕᵢⱼₖₗₘₙₒₚᵣₛₜᵤᵥ ₓ ᵦ
º¹²³⁴ⁿ₁₂₃₄·∶αβγδεζηθ ικ λ μ νξοπρστυφχψω
∽ ⊥ ∠ ⊙ ⊕ ⊗∈∩∪∑∫∞≡≠±≈＄㏒㎡㎥㎎㎏㎜
ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ
ħ∂φ
∈⊂∂Δ∇∀∃e̅Ζ͏͏͏͏͏͏ Z̅
▹◃ ∧†⨯∙↑↓
┘˩⌋⎦┙┚┛
