기본 페이지 포트폴리오 대한민국의 전통건축 중국과 일본의 전통건축 서유럽과 미국의 건축 국역 청오경 현대 우주론 대한민국의 산풍경 백두대간 종주산행 네팔 히말라야 트레킹 몽블랑 지역 트레킹 요세미티 캐년 등 티베트 실크로드 야생 생물 파노라마사진 갤러리 클래식 레코드 갤러리 AT 포럼 트레킹 정보 링크


 로그인  회원가입

Wave (2) The wave equation
    김관석  2024-03-04 17:06:11, 조회수 : 61
- Download #1 : Wa_Fig_2s.jpg (444.4 KB), Download : 0



 2 The wave equation

          2.1 Partial derivatives    
We use ordinary dervatives to find the slope of a line (𝑚 = 𝑑𝑦/𝑑𝑥) or to determine the speed of an object given its position as a function of time (𝑣𝑥 = 𝑑𝑥/𝑑𝑡). But because wavefuntions (𝑦) generally depend on two or more variables, such as distance (𝑥) and time (𝑡): 𝑦 = 𝑓(𝑥, 𝑡), we must use partial dervatives which are written as ∂/∂𝑥 or ∂/∂𝑡.
   Notice that at the location shown in Fig. 2.3 the slope of the surface is quite steep in the directiom of increasing 𝑥 (while remaining at the same value of 𝑡), but the slope is almost zeo if you move in the direction of increasing 𝑡 (while holding our 𝑥-value constant). We can see how this works in the following example.

Example 2.1 For the function 𝑦(𝑥, 𝑡) = 3𝑥2 - 5𝑡, find the partial derivative of 𝑦 with respect to 𝑥 and with respect to 𝑡.
      ∂𝑦/∂𝑥 = ∂(3𝑥2 - 5𝑡)/∂𝑥 = ∂(3𝑥2)/∂𝑥 - ∂(5𝑡)/∂𝑥 = 3∂(𝑥2)/∂𝑥 - 0 = 6𝑥.
      ∂𝑦/∂𝑡 = ∂(3𝑥2 - 5𝑡)/∂𝑡 = ∂(3𝑥2)/∂𝑡 - ∂(5𝑡)/∂𝑡 = 0 - 5∂𝑡/∂𝑡 = -5.

We can take higher-order partial derivaives. So for example
      ∂/∂𝑥 (∂𝑦/∂𝑥) = ∂2𝑦/∂𝑥2     ∂/∂𝑡 (∂𝑦/∂𝑡) = ∂2𝑦/∂𝑡2.
For example, consider the wave function 𝑦(𝑥, 𝑡) = 𝐴 sin(𝑘𝑥 - 𝜔𝑡) plotted in Fig. 2.4. In this plot, we can see the behavior of 𝑦 over distance and the behavior of 𝑦 over time. And the first partial derivative of 𝑦 with repect to 𝑥 is
    (2.1)   ∂𝑦/∂𝑥 = ∂[𝐴 sin(𝑘𝑥 - 𝜔𝑡)]/∂𝑥 = 𝐴 ∂[sin(𝑘𝑥 - 𝜔𝑡)]/∂𝑥 = 𝐴 cos(𝑘𝑥 - 𝜔𝑡) ∂(𝑘𝑥 - 𝜔𝑡)/∂𝑥 = 𝐴 cos(𝑘𝑥 - 𝜔𝑡) [∂(𝑘𝑥)/∂𝑥 - (𝜔𝑡)/∂𝑥] = 𝐴 cos(𝑘𝑥 - 𝜔𝑡) [𝑘∂𝑥/∂𝑥 - 0]     or      ∂(𝜔𝑡)/∂𝑥 = 𝐴𝑘 cos(𝑘𝑥 - 𝜔𝑡).
The second partial dervative of 𝑦 with respect to 𝑥 is
    (2.2)   ∂2𝑦/∂𝑥2 = ∂[𝐴𝑘 cos(𝑘𝑥 - 𝜔𝑡)]/∂𝑥 = 𝐴𝑘 ∂[cos(𝑘𝑥 - 𝜔𝑡)]/∂𝑥 = -𝐴𝑘 sin(𝑘𝑥 - 𝜔𝑡) ∂[(𝑘𝑥 - 𝜔𝑡)]/∂𝑥 = -𝐴 sin(𝑘𝑥 - 𝜔𝑡) [𝑘∂(𝑘𝑥)/∂𝑥 - (𝜔𝑡)/∂𝑥] = -𝐴𝑘 sin(𝑘𝑥 - 𝜔𝑡) [𝑘 ∂𝑥/∂𝑥 - 0]     or      ∂2𝑦/∂𝑥2 = -𝐴𝑘2 sin(𝑘𝑥 - 𝜔𝑡).
   Plots of the wavefunction and its first and second partial derivatives with respect to 𝑥 at time 𝑡 = 0 are shown in Fig. 2.5. If we are wondering how the cosine shape of the first partial derivatives of 𝑦(𝑥, 𝑡) relates to the shape of the wavefunction. And turning now to the behavior of this function over time, the first partial derivative of 𝑦 with respect to 𝑡 is
    (2.3)   ∂𝑦/∂𝑥 = ∂[𝐴 sin(𝑘𝑥 - 𝜔𝑡)]/∂𝑡 = 𝐴 ∂[sin(𝑘𝑥 - 𝜔𝑡)]/∂𝑡 = 𝐴 cos(𝑘𝑥 - 𝜔𝑡) ∂(𝑘𝑥 - 𝜔𝑡)/∂𝑡 = 𝐴 cos(𝑘𝑥 - 𝜔𝑡) [∂(𝑘𝑥)/∂𝑡 - (𝜔𝑡)/∂𝑡] = 𝐴 cos(𝑘𝑥 - 𝜔𝑡) [0 - 𝜔 ∂𝑡/∂𝑡]     or      ∂𝑦/∂𝑡 = -𝐴𝜔 cos(𝑘𝑥 - 𝜔𝑡).
The second partial dervative of 𝑦 with respect to 𝑡 is
    (2.4)   ∂2𝑦/∂𝑡2 = ∂[-𝐴𝜔 cos(𝑘𝑥 - 𝜔𝑡)]/∂𝑡 = -𝐴𝜔 ∂[cos(𝑘𝑥 - 𝜔𝑡)]/∂𝑡 = 𝐴𝜔 sin(𝑘𝑥 - 𝜔𝑡) ∂(𝑘𝑥 - 𝜔𝑡)/∂𝑡 = 𝐴𝜔 sin(𝑘𝑥 - 𝜔𝑡) [∂(𝑘𝑥)/∂𝑡 - (𝜔𝑡)/∂𝑡] = 𝐴𝜔 sin(𝑘𝑥 - 𝜔𝑡) [0 - 𝜔 ∂𝑡/∂𝑡]     or      ∂2𝑦/∂𝑡2 = -𝐴𝜔2 sin(𝑘𝑥 - 𝜔𝑡).

          2.2 The classical wave equation    
The most common form of the wave equation which is often called the "classical" wave equation looks like this:
    (2.5)   ∂2𝑦/∂𝑥2 = 1/𝑣22𝑦/∂𝑡2.
For example, consider 𝑦(𝑥, 𝑡) = 𝐴 sin(𝑘𝑥 - 𝜔𝑡) repesenting a sinusoidal wave traveling in the positive 𝑥-direction. We can start to compare the distance step to the time slope.
    (2.6)   𝛥𝑥 = 𝑣 𝛥𝑡.
Then the above equation can be written as 𝛥𝑦/𝛥𝑥 = 𝑣 𝛥𝑦/𝛥𝑡 and allowing the distance and time increments to shrink toward zero. Then the 𝛥𝑠 (𝛥𝑦/𝛥𝑥 or 𝛥𝑦/𝛥𝑡) become partial derivatives, and this relationship between slopes may be written as
    (2.7-8)   ∂𝑦/∂𝑡 = -𝑣 ∂𝑦/∂𝑥     or     ∂𝑦/∂𝑥 = -1/𝑣 ∂𝑦/∂𝑡;     ∂𝑦/∂𝑡 = 𝑣 ∂𝑦/∂𝑥     or     ∂𝑦/∂𝑥 = 1/𝑣 ∂𝑦/∂𝑡.
The first equation Eq. (2.6) is applied to waves moving in the positive 𝑥-direction and the next one is for the wave moving in the negative 𝑥-direction.
There is a straightfoward route to solve the first- and second- order wave equation. We can start from Eq. (2.3)  
           Eq. (2.3): ∂𝑦/∂𝑡 = -𝐴𝜔 cos(𝑘𝑥 - 𝜔𝑡)     𝐴 cos(𝑘𝑥 - 𝜔𝑡) = -1/𝜔 ∂𝑦/∂𝑡,
and then substituting this into Eq. (2.1):
           Eq. (2.1): ∂𝑦/∂𝑥 = 𝐴𝑘 cos(𝑘𝑥 - 𝜔𝑡) = -𝑘/𝜔 ∂𝑦/∂𝑡.
   Using the relation 𝑣 = 𝜔/𝑘 (Eq. (1.36)) makes this
           ∂𝑦/∂𝑥 = -1/𝑣 ∂𝑦/∂𝑡,
which is the first-order wave equation for was propagating in the positive 𝑥-dirction.
   Performing a similar analysis on the second-order equations (Eqs. (2.2) and (2.4)) yeilds the second the second-order classical wave equation.
           Eq. (2.4): ∂2𝑦/∂𝑡2 = -𝐴𝜔2 sin(𝑘𝑥 - 𝜔𝑡)     𝐴 sin(𝑘𝑥 - 𝜔𝑡) = - 1/𝜔22𝑦/∂𝑡2,
which we can then substitute into Eq. (2.2):
           ∂2𝑦/∂𝑥2 = -𝐴𝑘2 sin(𝑘𝑥 - 𝜔𝑡) = -𝑘2/𝜔22𝑦/∂𝑡2
Again using the relatiion 𝑣 = 𝜔/𝑘 (Eq. (1.36)) gives
           ∂2𝑦/∂𝑥2 = 1/𝑣22𝑦/∂𝑡2,
which is the classical second-order wave equation (Eq. (2.5)). Starting with the wavefunction 𝑦(𝑥, 𝑡) = 𝐴 sin(𝑘𝑥 + 𝜔𝑡) gives the same result.
   We may encounter versions of the wave equation that looks quite different from the presented in this section. One common way of writing Eq. (2.2) for harmonic wave is
    (2.9)   ∂2𝑦/∂𝑥2 = -𝐴𝑘2 sin(𝑘𝑥 - 𝜔𝑡) = -𝑘2𝑦,     or
    (2.10)   ∂2𝑦/∂𝑡2 = -𝐴𝜔2 sin(𝑘𝑥 - 𝜔𝑡) = -𝜔2𝑦,
since 𝑦 = 𝐴2 sin(𝑘𝑥 - 𝜔𝑡) in this case.
   We may also encoiunter the "dot" and "double-dot" notation, in which first derivative with respect to time are signified by a dot over the variable and second derivatives with respect to time are signified by two dots over the variable. Using this notation Eq. (2.10) becomes
          ÿ= -𝜔2𝑦     and      ∂2𝑦/∂𝑥2 = 1/𝑣2 ÿ.  
   Another common notation for derivatives uses subscripts to indicate the variable with respect to which the partial derivatives is taken. For example.
          ∂𝑦/∂𝑥 ≡ 𝑦𝑥,
where the symbol ≡ means "is defined as". The second partial derivative with respect to 𝑡 may be written as
          ∂2𝑦/∂𝑡2 ≡ 𝑦𝑡𝑡,
so the classical wave equation looks like this:
          𝑦𝑥𝑥 = 1/𝑣2 𝑦𝑡𝑡.
   The classical wave equation can be expanded to higher dimensions by adding partial derivatives in other directions. For example, for a spatially three-dimensional wavefunction Ψ(𝑥, 𝑦, 𝑧, 𝑡), the classical wave equation is
    (2.11)   ∂2Ψ/∂𝑥2 + ∂2Ψ/∂𝑦2 + ∂2Ψ/∂𝑧2 = 1/𝑣22Ψ/∂𝑡2.
We may see this written as
    (2.12)   𝛻2Ψ = 1/𝑣22Ψ/∂𝑡2,
in which the symbol 𝛻2 represents the Laplacian operator.

          2.3 Properties of the wave equation    

Partial. The classical wave equation is a partial differential equation (PDE), because it depends on changes in the wavefunction with respect to more than one variables (such as 𝑥 and 𝑡). The alternative is an ''ordinary'' differential operation (ODE), which dipends on changes with respect to only a single variable. An example of the latter is Newton's second law, which staes that the acceleration of an object is equal to the sum of the external forces on the object (∑𝐹ext) divided by the object's mass (𝑚). The one-dimensional version is
    (2.13)   𝑑2𝑥/𝑑𝑡2 = ∑𝐹ext/𝑚.
We can see how the PDE works in Section 2.4 of this chapter.

Homogeneous. The classical wave equation is homogeneous because it contains only terms involve the dependent variable or derivatives of the dependent variable. Mathematically, that means the classical (homogeneous) wave equation looks like
    (2.14)   ∂2𝑦/∂𝑥2 - 1/𝑣22𝑦/∂𝑡2 = 0.
as opposed to the inhomogeneous case, which looks like
    (2.15)   ∂2𝑦/∂𝑥2 - 1/𝑣22𝑦/∂𝑡2 = 𝐹(𝑥, 𝑡),
where 𝐹(𝑥, 𝑡) represents some function of the indepenent variables 𝑥 and 𝑡 (but not 𝑦). 
   The external function 𝐹(𝑥, 𝑡) is called a "source" or "an "external force", and those are good clue as it meaning.Whatever they're called, The term not involving the dependent variable always represents an external stimulus of some kind. To see that, look back at Eq. (2.13).

Second-order. The classical wave equation is a second-order partial diffrential equation. It's the change in the waveform's slope over distance that's related to the change in the slope over time are second derivatives.

Hyperbolic. We may encounter the wave equation called a "hyperbolic" differential equation. Hyperbolas are a form of conic section (along with ellipses and parabolas) that can be represented by simple equations. The classical wave equation has a similar form to the equation for a hyperbola:
    (2.16)   𝑦2/𝑎2 - 𝑥2/𝑏2 = 1,
in which the constants 𝑎 and 𝑏 determine the "flatness" of the hyperbola.
   To compare this with the classical wave equation (Eq. (2.5)), it helps to first get both terms onto the left side:
    (2.17)   ∂2𝑦/∂𝑥2 - 1/𝑣22𝑦/∂𝑡2 = 0,
where the scond derivatives as well as the negative sign between them make the wave equation "hyperbolic". If we consider the equation
    (2.18)   𝑦2/𝑎2 - 𝑥2/𝑏2 = 0,
we can find that the solutions are two straight lines that cross at the origin which is a special case of a hyperbola. And, we will in the section 2.4, there are useful differential equations with a first-order time derivative and a second-order space derivative, and such equations are characterized as "parabolic".

Linear. The classical wave equation is linear because all of the terms involving the wavefunction 𝑦(𝑥, 𝑡) and derivatives of 𝑦(𝑥, 𝑡) are raised to the first power, and there are no cross terms involving the product of the wavefunction and its dervatives. If a differential equation does include terms which higher powers or cross terms of the wavefunction and its derivatives, that diffrential equation is said to be nonlinear.
   An extremely powerful characteristic of all linear differential equations is that solutions obey the "superposition principle". Mathematically, the superposition principle says that, if two wavefunction 𝑦1(𝑥, 𝑡) and 𝑦2(𝑥, 𝑡) are each a solution to the linear equation, then their sum at every point in space and time, 𝑦total(𝑥, 𝑡) = 𝑦1(𝑥, 𝑡) + 𝑦2(𝑥, 𝑡). We can prove this for two waves traveling at the same speed 𝑣 by writing the wave equation of each wave:
    (2.19)   ∂2𝑦1(𝑥, 𝑡)/∂𝑥2 - 1/𝑣22𝑦1(𝑥, 𝑡)/∂𝑡2 = 0,     ∂2𝑦2(𝑥, 𝑡)/∂𝑥2 - 1/𝑣22𝑦2(𝑥, 𝑡)/∂𝑡2 = 0,
and then adding these two equations:
            ∂2𝑦1(𝑥, 𝑡)/∂𝑥2 + ∂2𝑦2(𝑥, 𝑡)/∂𝑥2 - (1/𝑣2) ∂2𝑦1(𝑥, 𝑡)/∂𝑡2 - (1/𝑣2) ∂2𝑦2(𝑥, 𝑡)/∂𝑡2 = 0.
This can be simplified to
            ∂2[𝑦1(𝑥, 𝑡) + 𝑦2(𝑥, 𝑡)]/∂𝑥2 - 1/𝑣22[𝑦1(𝑥, 𝑡) + 𝑦2(𝑥, 𝑡)]/∂𝑡2 = 0
and, since 𝑦total(𝑥, 𝑡) = 𝑦1(𝑥, 𝑡) + 𝑦2(𝑥, 𝑡), this is
    (2.20)   ∂2𝑦total(𝑥, 𝑡)/∂𝑥2 - 1/𝑣22𝑦total(𝑥, 𝑡)/∂𝑡2 = 0.

Example 2.2 Consider two sine waves with following wavefunctions:
            𝑦1(𝑥, 𝑡) = 𝐴1 sin(𝑘1𝑥 + 𝜔1𝑡 + 𝜖1), 
            𝑦2(𝑥, 𝑡) = 𝐴2 sin(𝑘2𝑥 + 𝜔2𝑡 + 𝜖2).
If 𝐴1 = 𝐴2 = 1, 𝑘1 = 𝑘2 = 1, 𝜔1 = 𝜔2 = 2 rad/s, but 𝜖1 = 0 and 𝜖2 = π/3, determine the characteristics of the wave that results from the addition of these waves.
    (2.21)   𝑦1(𝑥, 𝑡) = 𝐴1sin(𝑘1𝑥 + 𝜔1𝑡 + 𝜖1) = sin(𝑥 + 2𝑡 + 0), 
            𝑦2(𝑥, 𝑡) = 𝐴2sin(𝑘2𝑥 + 𝜔2𝑡 + 𝜖2) = sin(𝑥 + 2𝑡 + π/3),
    (2.22)   𝑦total(𝑥, 𝑡) = sin(𝑥 + 2𝑡) + sin(𝑥 + 2𝑡 + π/3),
    (2.23)   sin(𝜃1 + 𝜃2) = 2 sin[(𝜃1 + 𝜃2)/2]cos[(𝜃1 - 𝜃2)/2].
and plugging in 𝜃1 = 𝑥 + 2𝑡 and 𝜃2 = 𝑥 + 2𝑡 + π/3 gives
    (2.24)   𝑦total(𝑥, 𝑡) = 2 sin[(2𝑥 + 4𝑡 + π/3)/2]cos[-(π/3)/2].
where 𝐴 = 2cos(-π/6) ≈ 1.73, and 𝑥 = 0, they look like Fig. 2.13.
   Another very powerful way to analyze the superposition of waves is through the use of phasors. As shown in Fig. 2.14 for the location 𝑥 = 0 at time 𝑡 = 0, these example can be represented bu a rotating phasors. So we have  
            𝑦1(0, 0) = 𝐴1sin(𝑘1𝑥 + 𝜔1𝑡 + 𝜖1) = (1)sin(1(0) + 2(0) + 0) = 0, 
            𝑦2(𝑥, 𝑡) = 𝐴2sin(𝑘2𝑥 + 𝜔2𝑡 + 𝜖2) = (1)sin((1)(0) + 2(0) + π/3) ≈ 0.866.
  If we prefer to use the "addition of componets" approach to finding the resutant phasor 𝑦total, we can use the geometry of Fig. 2.14. Thus the magnitude and phase angle of 𝑦total are
    (2.25-26)   𝐴total = √(1.52 + 0.8662) = 1.73,     𝜖total= tan-1 (0.866/1.5) = π/6.

Summary. All of the characteristics of the classicasl wave equation discussed in this section are summerized in the expanded equation shown in Fig. 2.16

          2.4 PDEs related to wave equation    
There are other partial differential equations that pertain to motion through space and time, and those equations have some characteristics that are siiimilar and some that are different from the chracteristics of the classical wave equation.

The advection equation. In fact, we've already seen it in Section 2.2: It's the one-way equation:
    (2.27)   ∂𝑦(𝑥, 𝑡)/∂𝑥 = -1/𝑣 ∂𝑦(𝑥, 𝑡)/∂𝑡.
"Advection" is a kind of transport mechanism specially describing the way substances move when they are carried along in a current. For example pollutants in a river or pollen in an air current, the advection equation can model the movement more simply than the classical wave equation.

The Korteweg-de Vries equation. Not all wave equations are linear; a well-known example of a nonlinear wave equation is the Korteweg-d Viries (KdV) equation, which describes, for example, small-amplitude, shallow, and confined water waves, callled "solitary waves" or "solitons". A soliton looks like a wave pulse that retains its shape as it travels. The KdV equation has the form
    (2.28)   ∂𝑦(𝑥, 𝑡)/∂𝑡 - 6𝑦(𝑥, 𝑡)∂𝑦(𝑥, 𝑡)/∂𝑥 + ∂3𝑦(𝑥, 𝑡)/∂𝑥3 = 0.
The nonlinear term is the middle one, 6𝑦(∂𝑦/∂𝑥), beacause it includes the product of two different terms involing 𝑦(𝑥, 𝑡). The solutions no longer obey the superposition principle. However, once the solitons have passed through each other , they return to their original shapes.

The heat equation. While the classical wave equation is hyperbolic, the heat equation is parabolic, having a form analogous to 𝑦(𝑥, 𝑡) = 𝑎2. In other words, while it's still second-order in space, it's only first-order in time:
    (2.29)   ∂𝑦(𝑥, 𝑡)/∂𝑡 = 𝑎 ∂2𝑦(𝑥, 𝑡)/∂𝑥2,
where 𝑎 is the thermal diffusivity, a measure of how easily heat transfer through a system. The heat equation isn't classified a s a wave equation, because it is dissipative rather than oscillatory. By examining the time-dependent portion of the solution, which you can do through a common method called "separation of variables". The assumption behind this method is that, althogh the solution 𝑦(𝑥, 𝑡) depends on both space and time, the behavior in time 𝛵(𝑡) is independent of the behavor in space 𝑋(𝑥). That is, the wavefunction can be written as the product of 𝛵(𝑡) and 𝑋(𝑥):
    (2.30)   𝑦(𝑥, 𝑡)/∂𝑡 = 𝛵(𝑡)𝑋(𝑥).
   To see how that works, plug Eq. (2.30) into the jeat equation:
    (2.31)   ∂[𝛵(𝑡)𝑋(𝑥)]/∂𝑡 = 𝑎∂2[𝛵(𝑡)𝑋(𝑥)]/∂𝑥2.
Because 𝑋(𝑥) is a constant with respect to time and 𝛵(𝑡) is independent of space. Pulling these functions out of the derivatives leavers
    (2.32)   𝑋(𝑥)∂𝛵(𝑡)/∂𝑡 = 𝑎𝛵(𝑡)∂2𝑋(𝑥)/∂𝑥2.
The next step is to isolate all functions and derivatives of 𝑡 on one side and 𝑥 on the other. The heat equatiom now looks like
    (2.33)   1/𝛵(𝑡) ∂𝛵(𝑡)/∂𝑡 = 𝑎 1/𝑋(𝑥) ∂2𝑋(𝑥)/∂𝑥2.
The left side depends only on location and does not vary with time and the right side depends only on location and does not vary with time. But if this equation is true at every location at every time, then neither side can vary at all. Thus both sides must be constant, and since the sides equal one another, they must equal the same constant. If we call that constant -𝑏, then Eq. (2.33) becomes
    (2.34)   1/𝛵(𝑡) 𝑑𝛵(𝑡)/𝑑𝑡 = -𝑏,
which is an ordinary differential equation. Writing 𝛵(𝑡) as 𝛵 for simplicity and multiplying both sides by 𝑑𝑡 and integrating both sides gives
            1/𝛵 𝑑𝛵 = -𝑏 𝑑𝑡     𝑑𝛵/𝛵 = -𝑏 𝑑𝑡     or     ln 𝛵 = -𝑏𝑡 + 𝑐,
in which 𝑐 is the combind integration constant for both sides. If we apply the inverse function of ln, the natural log, which is the exponenyial: 𝑒ln 𝑇 = 𝑇. Doing so gives
    (2.35)   𝛵(𝑡) = 𝑒-𝑏𝑡+𝑐 = 𝑒-𝑏𝑡𝑒𝑐 = 𝐴𝑒-𝑏𝑡.
In this expression, the constant term 𝑒𝑐 has been absorbed into 𝐴. It's the last term (𝑒-𝑏𝑡) that make the solution dissipative.

The Scrödinger equation. The Scrödinger equation bears a stronger resemblance to the heat equation than to the classical wave equation, but its solution definitely have the characteristics of waves. Like the heat equation, the Scrödinger equation has a first-order derivative with repect to time , and a second-order derivative with respect to posirion. However, it has an additional factor of 𝑖 with the time derivative, and taht factor has a significant impact on the nature of the solutions. We can see that by considering this form of the Scrödinger equation:
    (2.36)   𝑖ℏ ∂𝑦(𝑥, 𝑡)/∂𝑡 = - ℏ2/2𝑚 ∂2𝑦(𝑥, 𝑡)/∂𝑡2 + 𝑉𝑦(𝑥, 𝑡),
where 𝑉 is the potential energy of the system and ℏ is the reduced Planck constant.
   Just as with the heat equation, the time behavior of the solutions can be found by using the separation of variables. Assuming that the overall solution has the form of Eq. (2.30), the Scrödinger equation becomes
    (2.37)   𝑖ℏ ∂𝛵(𝑡)𝑋(𝑥)/∂𝑡 = -ℏ2/2𝑚 ∂2𝛵(𝑡)𝑋(𝑥)/∂𝑡2 + 𝑉𝛵(𝑡)𝑋(𝑥).
Pulling 𝛵(𝑡) out of space derivatives and 𝑋(𝑥) out of the time derivatives and dividing by 𝛵(𝑡)𝑋(𝑥) gives
    (2.38)   𝑖ℏ/𝛵(𝑡) ∂𝛵(𝑡)/∂𝑡 = -ℏ2/2𝑚𝑋(𝑥) ∂2𝑋(𝑥)/∂𝑡2 + 𝑉.
Using the same reasoning as described above gives a time-only equation of
    (2.39)   𝑖ℏ/𝛵(𝑡) ∂𝛵(𝑡)/𝑑𝑡 = 𝐸.
This constant is the energy of the state, which is why it's called 𝐸. This is now an ordinary differential equation and can be arranged into
            𝑑𝛵/𝛵 = -𝑖𝐸/ℏ 𝑑𝑡.    
Now integrate each side,
            ln 𝛵 = -𝑖𝐸𝑡/ℏ + 𝑐,
and solve for 𝛵(𝑡):
    (2.40)   𝛵(𝑡) = 𝐴𝑒-𝑖𝐸𝑡/ℏ
Unlike the decaying exponetial function 𝑒-𝑥, the real and imaginary parts of 𝑒𝑖𝑥 are oscillatory.
                                                                        
* Textbook: D. Fleisch & J. Kinnaman A Student's Guide to Waves (Cambridge University Press 2015]


Name
Spamfree

     여기를 클릭해 주세요.

Password
Comment

  답글쓰기   목록보기
번호 제               목 이름 연관 날짜 조회
공지  '현대 우주론'에 관한 탐구의 장    관리자 1 2017-08-15
11:36:55
1246
공지  위키백과 업데이트: 로저 펜로즈  ✅   [1]  김관석 1 2021-09-28
06:56:21
2046
127  Wave (1) Wave fundamentals    김관석 8 2024-05-07
09:24:42
406
   Wave (2) The wave equation    김관석 8 2024-05-07
09:24:42
406
125      Wave (3a) General solution; Boundary conditions    김관석 8 2024-05-07
09:24:42
406
124        Wave (3b) Fourier theory    김관석 8 2024-05-07
09:24:42
406
123          Wave (3c) Wave packets and dispersion    김관석 8 2024-05-07
09:24:42
406
122            Wave (4) Mechanical wave equation  ✍️ (proofreading)    김관석 8 2024-05-07
09:24:42
406
121              Wave (5) Electromagnetic wave equation  ✍️ (proofreading)    김관석 8 2024-05-07
09:24:42
406
120                Wave (6) Quantum wave equation  ✍️ (proofreading)    김관석 8 2024-05-07
09:24:42
406
119  Gravirational Collapse and Space-Time Singuarities  📚    김관석 1 2023-06-12
13:50:03
152
118  Mathematics of Astronomy  (1) Gravity; Light    김관석 4 2023-05-02
08:31:40
907
117    Mathematics of Astronomy  (2) Parallax, angular size etc.      김관석 4 2023-05-02
08:31:40
907
116      Mathematics of Astronomy  (3) Stars    김관석 4 2023-05-02
08:31:40
907
115        Mathematics of Astronomy  (4) Black holes & cosmology    김관석 4 2023-05-02
08:31:40
907
114   Still의 <블록으로 설명하는 입자물리학>      김관석 3 2022-04-14
18:49:01
848
113    Becker의 <실재란 무엇인가?>    김관석 3 2022-04-14
18:49:01
848
112      Penrose의 <시간의 순환> (강추!) [u. 5/2023]  🌹    김관석 3 2022-04-14
18:49:01
848
111  일반상대성(GR) 학습에 대하여..    김관석 1 2022-01-03
09:49:28
303
110  HTML(+) 리뷰/홈페이지 운용^^  [1]  김관석 1 2021-11-08
16:52:09
205
109  Peebles의 Cosmology's Century (2020)    김관석 1 2021-08-16
21:08:03
386
108  <한권으로 충분한 우주론> 외  ✅    김관석 5 2021-06-06
13:38:14
2056
107    Rovelli의 <보이는 세상은 실재가 아니다>    김관석 5 2021-06-06
13:38:14
2056
106      Smolin의 <양자 중력의 세가지 길>    김관석 5 2021-06-06
13:38:14
2056
105        Susskind의 <우주의 풍경> (강추!)  🌹    김관석 5 2021-06-06
13:38:14
2056
104          대중적 우주론 추천서 목록 [u. 9/2021]  [1]  김관석 5 2021-06-06
13:38:14
2056
103  Zel'dovich's Relativistic Astrophysics  ✅    김관석 1 2021-04-01
08:16:42
1201
102  Dirac Equation and Antimatter    김관석 1 2021-03-15
12:49:45
532
101  11/30 태양 흑점 sunspots    김관석 2 2020-11-30
16:14:27
965
100    Coronado PST 태양 사진^^    김관석 2 2020-11-30
16:14:27
965
99  Linde's Inflationary Cosmology [u. 1/2021]    김관석 1 2020-11-06
09:19:06
530
98  The Schrödinger Equation [완료] (7) Harmonic Oscillator  ✅    김관석 1 2020-09-17
21:43:31
2686
97  우주론의 명저 Weinberg의 <최초의 3분>  ✅    김관석 3 2020-08-09
11:37:44
1334
96    물리학도를 위한 우주론서는?    김관석 3 2020-08-09
11:37:44
1334
95      우주론의 최고, 최신, 고전서..    김관석 3 2020-08-09
11:37:44
1334
94   Mathematical Cosmology I. Overview  🔵    김관석 6 2020-06-07
16:23:00
4438

    목록보기   다음페이지     글쓰기 1 [2][3][4]
    

Copyright 1999-2024 Zeroboard / skin by zero & Artech