User:Justin545/Private Note

From Wikipedia, the free encyclopedia

1 Function Expansion, Basis Function of Dirac Delta
2 Quantum: Measurement vs. Schrödinger Equation
3 Data Compression by Specifying The Space and Time
4 Quantum Mechanics: Determine The Electric/Potential Field of a Fermion
5 Quantum Viewpoint: Optic Filter & Measurement
6 Quantum: Does an Electron Occupies Space?
7 (temporary)
- 7.1 Proof
  - 7.1.1 All of eigenvalues are real
  - 7.1.2 Orthogonality of the eigenfunctions
- 7.2 Format
8 south park

[edit] Function Expansion, Basis Function of Dirac Delta

Dirac delta $δ(t)$ can be viewed as the limit of

$\delta(t)=\lim_{a\to 0}\delta_a(t)$		(1)

where $δ a (t)$ is sometimes called a nascent delta function. There are many kind of definitions of $δ a (t)$ . For example, it can be defined as

$\delta_a(t)=\begin{cases} \frac{1}{a},&-\frac{a}{2}<t<\frac{a}{2}\\ 0,&\mbox{otherwise} \end{cases}$		(2)

Using Fourier transforms, one finds that

$\int_{-\infty}^\infty 1 \cdot e^{-i 2\pi f t}\,dt = \delta(f)$		(3)

and therefore

$\int_{-\infty}^\infty e^{i 2\pi f_1 t} \left(e^{i 2\pi f_2 t}\right)^*\,dt = \int_{-\infty}^\infty e^{-i 2\pi (f_2 - f_1) t} \,dt = \delta(f_2 - f_1)$		(4)

which is a statement of the orthogonality property for the Fourier kernel.

Similarly, we can show the orthogonality property for the Dirac delta. Consider the property of Dirac delta

$\int_{-\infty}^\infty f(t)\delta(t-T)\,dt=f(T)$		(5)

Replace $T$ by $x$ in (5)

$\int_{-\infty}^\infty f(t)\delta(t-x)\,dt=f(x)$		(6)

Let

$f(t)\equiv\delta(t-y)$		(7)

Replace $t$ by $x$ in (7)

$f (x) = δ(x - y)$		(8)

Replace (7) and (8) into (6)

$\int_{-\infty}^\infty\delta(t-y)\delta(t-x)\,dt=\delta(x-y)$		(9)

$\int_{-\infty}^\infty\delta(t-x)\delta(t-y)\,dt=\delta(x-y)$		(10)

Thus, Dirac delta are orthogonal eigenfunctions. According to Sturm-Liouville theory, a given function $f (t)$ , satisfying suitable conditions, can be expanded in an infinite series of eigenfunctions $φ n (t)$ of the more general Sturm–Liouville problem of

$\left[p(x)y'\right]'-q(x)y+\lambda r(x)y=0$		(11)

$a 1 y (0) + a 2 y'(0) = 0$		(12)

$b 1 y (1) + b 2 y'(1) = 0$		(13)

such that

$f(t)=\sum_{n=1}^\infty c_n\phi_n(t)$		(14)

Each element of the set of eigenfunctions $\{\phi_n(t)\}_{n=1}^\infty$ is a solution satisfying the more general Sturm–Liouville problem (11), (12) and (13).

could be expressed as a series of eigenfunctions  $φ n (t)$  such that

as a series of eigenfunctions of Dirac delta?

p.s. I don't know how to classify this question, so I put it here rather than Wikipedia:Reference_desk/Science because I think more math is involved than quantum mechanics. - Justin545 (talk) 05:24, 24 March 2008 (UTC)

[edit] Quantum: Measurement vs. Schrödinger Equation

1. Article Copenhagen interpretation: Each measurement causes a change in the state of the particle, known as wavefunction collapse.

2. Article Schrödinger equation: The Schrödinger equation is commonly written as an operator equation describing how the state vector evolves over time.

Although I don't fully understand quantum mechanics, the two items above seem to be related to each other.

When an observable of a quantum system is measured, the state $|\psi(t)\rangle$ of the system can be expressed as

$\|\psi(t)\rangle=\sum_i\psi_i\|i\rangle$		(1)

where $|i\rangle$ is the

i

th eigenfunction, which is associated to eigenvalue

i

, of the observable and

$\psi_i=\langle i\|\psi(t)\rangle$		(2)

which will "suddenly" or "discretely" collapse from $|\psi\rangle$ to one of terms, say $\psi_a|a\rangle$ , of the right-hand side of (1). The rest of the terms not associated to eigenvalue $a$ simply vanish after the measurement.

On the other hand, Schrödinger equation

$\hat H(t)\left\|\psi\left(t\right)\right\rangle = \mathrm{i}\hbar \frac{d}{d t} \left\| \psi \left(t\right) \right\rangle$		(3)

where

$\hat H(t)=-\frac{\hbar^2}{2m}\nabla^2+V\left(\mathbf{r},t\right)$		(4)

describing how the state vector $|\psi(t)\rangle$ evolves over time. When the state $|\psi(t)\rangle$ of the system is measured, the apparatus measuring the system will interact with the system and makes change to the potential field $V\left(\mathbf{r},t\right)$ . Therefore, the state $|\psi(t)\rangle$ should evolve "smoothly" or "continuously" according to the varying potential $V\left(\mathbf{r},t\right)$ during the measurement. According to Schrödinger Equation (3) and (4) together with $V\left(\mathbf{r},t\right)$ , we should be able to figure out the final state of the system after the measurement.

It seems that the measuring process can be explained by the two ways, wavefunction collapse & Schrödinger equation, above. Do they contradict? Is "wavefunction collapse" compatible with "Schrödinger Equation"? - Justin545 (talk) 08:12, 26 March 2008 (UTC)

[edit] Data Compression by Specifying The Space and Time

cosmology, big bang, initial state, absolute time, relativity - Justin545 (talk) 03:17, 26 March 2008 (UTC)

[edit] Quantum Mechanics: Determine The Electric/Potential Field of a Fermion

[edit] Quantum Viewpoint: Optic Filter & Measurement

optic filter - Justin545 (talk) 08:54, 19 April 2008 (UTC)

[edit] Quantum: Does an Electron Occupies Space?

Does an electron has volume? What is the diameter of an electron? Or an electron is just a imaginary (ideal) point in space and does not occupy space? Or an electron is a particle consisted of subatomic particles which do not occupy space?

If an electron occupies space, what will happen if two electrons collide? Will they just overlap with each other and then pass through without collision?

[edit] (temporary)

[edit] Proof

$-\frac{d}{dx}\left[p(x)\frac{dy}{ dx}\right]+q(x)y=\lambda w(x)y.$		(1)

$y(a)\cos \alpha - p(a)y^{\prime}(a)\sin \alpha = 0,$		(2)

$y(b)\cos \beta - p(b)y^{\prime}(b)\sin \beta = 0,$		(3)

$L u =-{d\over dx}\left[p(x){du\over dx}\right]+q(x)u$		(xx4)

Rearrange terms of (2) and (3), we have

$y'(a)=\frac{y(a)\cos\alpha}{p(a)\sin\alpha}$		(?)

$y'(b)=\frac{y(b)\cos\beta}{p(b)\sin\beta}$		(6)

If the boundary conditions is

a 1 y (0) + a 2 y'(0) = 0

b 1 y (1) + b 2 y'(1) = 0

Rearrange terms, we get

$y'(0)=-\frac{a_1}{a_2}y(0)$		(1)

$y'(1)=-\frac{b_1}{b_2}y(1)$		(1)

$y_1'(b)=\frac{y_1(b)\cos\beta}{p(b)\sin\beta}$

$y_2'(b)=\frac{y_2(b)\cos\beta}{p(b)\sin\beta}$

$y_1'(a)=\frac{y_1(a)\cos\alpha}{p(a)\sin\alpha}$

$y_2'(a)=\frac{y_2(a)\cos\alpha}{p(a)\sin\alpha}$

L y 1 = λ 1 w (x) y 1

L y 2 = λ 2 w (x) y 2

$\int_c^d(Lu)v-u(Lv)\,dx=-p(u'v-uv')\bigg\|_c^d$		(6)

$\int_a^b(Ly_1)y_2-y_1(Ly_2)\,dx=-p(y_1'y_2-y_1y_2')\bigg|_a^b$

$\int_a^b[Ly_1(x)]y_2(x)-y_1(x)[Ly_2(x)]\,dx=-p(x)[y_1'(x)y_2(x)-y_1(x)y_2'(x)]\bigg|_a^b$

$\int_a^b[Ly_1(x)]y_2(x)-y_1(x)[Ly_2(x)]\,dx=-p(b)\left[\frac{y_1(b)\cos\beta}{p(b)\sin\beta}y_2(b)-y_1(b)\frac{y_2(b)\cos\beta}{p(b)\sin\beta}\right]+p(a)\left[\frac{y_1(a)\cos\alpha}{p(a)\sin\alpha}y_2(a)-y_1(a)\frac{y_2(a)\cos\alpha}{p(a)\sin\alpha}\right]$

$\int_a^b[Ly_1(x)]y_2(x)-y_1(x)[Ly_2(x)]\,dx=-p(b)\left[\frac{\cos\beta}{p(b)\sin\beta}y_1(b)y_2(b)-y_1(b)y_2(b)\frac{\cos\beta}{p(b)\sin\beta}\right]+p(a)\left[\frac{\cos\alpha}{p(a)\sin\alpha}y_1(a)y_2(a)-y_1(a)y_2(a)\frac{\cos\alpha}{p(a)\sin\alpha}\right]$

In order to render the theory as simple as possible while retaining considerable generality, we assume w(x) is a real-valued function and w(x) > 0 for all x on the interval [a,b].^[1] In terms of linear boundary value problems, Lagrange's identity is

$\int_0^1(Lu)v-u(Lv)\,dx=-p(u'v-uv')\bigg|_0^1$

Retrace the steps in the proof of Lagrange's identity, we can also proof that

$\int_c^d(Lu)v-u(Lv)\,dx=-p(u'v-uv')\bigg\|_c^d$		(6)

If v=u, the identity (6) becomes

$\int_c^d(Lu)u-u(Lu)\,dx=-p(u'u-uu')\bigg|_c^d$

$\int_c^d(Lu)u-u(Lu)\,dx=-p\cdot 0\bigg|_c^d=0$

$\int_c^d(Lu)u\,dx=\int_c^du(Lu)\,dx$		(7)

Actually, it can be shown that the identity (7) becomes

$\int_c^d(Lu)\overline u\,dx=\int_c^du\overline{(Lu)}\,dx$		(8)

when u is a complex-valued function of x.^[1] Whereas the overlines denote the complex conjugate. Suppose λ_n is the n-th eigenvalue of the problem (1)-(2)-(3) and y_n is the corresponding eigenfunction. Because λ_n and y_n are possibly complex-valued, we presume that they have the forms λ_n = A + iB and y_n = C(x) + iD(x), where A, B, C(x) and D(x) are real. Replace c=a, d=b and u=y_n into (8), we have

$\int_a^b(Ly_n)\overline{y_n}\,dx=\int_a^by_n\overline{(Ly_n)}\,dx$		(9)

Replace u=y_n into (xx4), we have

$Ly_n=-{d\over dx}\left[p(x){dy_n\over dx}\right]+q(x)y_n$		(10)

Since y_n is an eigenfunction, it also satisfies (1), that is

$-\frac d{dx}\left[p(x)\frac{dy_n}{dx}\right]+q(x)y_n=\lambda_n w(x)y_n$		(11)

Replace (11) into (10), we have

$L y n = λ n w (x) y n$		(12)

[edit] All of eigenvalues are real

Replace (12) into (9), we have

$\int_a^b[\lambda_n w(x)y_n]\overline{y_n}\,dx=\int_a^by_n\overline{[\lambda_n w(x)y_n]}\,dx$

$\int_a^b\lambda_n w(x)y_n\overline{y_n}\,dx=\int_a^by_n\overline{\lambda_n}\,\overline{w(x)}\overline{y_n}\,dx$		(13)

Since w(x) is real, (13) becomes

$\int_a^b\lambda_n w(x)y_n\overline{y_n}\,dx=\int_a^by_n\overline{\lambda_n}w(x)\overline{y_n}\,dx$		(14)

Rearrange terms of (14), we have

$(\lambda_n-\overline{\lambda_n})\int_a^bw(x)y_n\overline{y_n}\,dx=0$

$(\lambda_n-\overline{\lambda_n})\int_a^bw(x)[C^2(x)+D^2(x)]\,dx=0$		(15)

Because all eigenfunctions are not trivial solution which means y_n ≠ 0. Also w(x) > 0, we conclude the integration part of (15) is not zero and

$\lambda_n-\overline{\lambda_n}=0$

(A + i B) - (A - i B) = 2 i B = 0

which means B = 0. So the eigenvalue λ_n is real. Q.E.D.

[edit] Orthogonality of the eigenfunctions

^ ^a ^b Boyce, William E.; Richard C. DiPrima (2001). "Boundary Value Problems and Sturm–Liouville Theory", Elementary Differential Equations and Boundary Value Problems (PDF), 7th ed. (in English), New York: John Wiley & Sons, pp. 630-632. ISBN 0-471-31999-6. OCLC 64431691. “We assume that the functions p, p', q, and r are continuous on the interval 0 ≤ x ≤ 1 and, further, that p(x) > 0 and r(x) > 0 at all points in 0 ≤ x ≤ 1. These assumptions are necessary to render the theory as simple as possible while retaining considerable generality. ... It is important to know that Eq. (8) remains valid under the stated conditions if u and v are complex-valued functions and if the inner product (9) is used. ... Since r(x) is real, Eq. (13) reduces to ...”

[edit] Format

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[edit] south park

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   [____________] /` |________| `\  /   .'``'.   \    /_.-"`_  _`"-._\
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   s'----------'    '----------'   '--------------'`--------------------`
      S at N          K Y L E        K E N N Y         C A R T M A N
omg it's south park
omg no f-ing kidding.