Wave mechanics - Revision history

Milllo: /* Introduction */

2020-05-27T06:26:09Z

Introduction

2020-05-27T06:17:14Z

Solutions to Wave Equation in One Dimension

2020-05-27T06:15:25Z

Solutions to Wave Equation in One Dimension

2020-05-06T00:02:37Z

Wave Equation in One Dimension

2020-05-06T00:00:28Z

Second Partial Derivatives of basic wave equation

2020-05-05T23:52:45Z

Basic Equation

2020-05-05T23:52:20Z

Second Partial Derivatives of basic wave equation

2020-05-05T23:50:42Z

Partial Derivatives of basic wave equation

2020-05-05T23:48:08Z

Introduction

2020-05-05T19:31:12Z

Solutions to Wave Equation in One Dimension

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 ==Introduction==
-Wave mechanics studies the properties of wave functions. These are periodic functions that carry information due to their periodic nature. The best way to study a function is to see how it changes, which can be done using derivatives, or derivatives of derivatives.
+Wave mechanics studies the properties of wave functions. These are periodic functions that carry information due to their periodic nature. The best way to study a function is to see how it changes, which can be done using derivatives, or derivatives of derivatives. Wave mechanics are general and need have no physical interpretation. Real systems that behave as waves can be modeled with wave mechanics, but as with all real systems, there are complexities that are not captured by the math. These complexities can be approximated using [[Perturbation Theory]].
 ==Basic Equation==

← Older revision		Revision as of 06:17, 27 May 2020
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	where F and G are smooth functions.		where F and G are smooth functions.
		+
		+	We already know that y(x,t) = Ψ = Asin(kx - ωt) is a solution to the wave equation since this is the equation that we used to derive it. Thus there are many solutions but they all fit the form y(x,t) = F(x-vt) + G(x+vt).

← Older revision		Revision as of 06:15, 27 May 2020
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	y(x,t) = F(x-vt) + G(x+vt)		y(x,t) = F(x-vt) + G(x+vt)
		+
		+	where F and G are smooth functions.

← Older revision		Revision as of 00:02, 6 May 2020
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	v<sup>2</sup> ∂<sup>2</sup>y/∂x<sup>2</sup> = ∂<sup>2</sup>y/∂t<sup>2</sup>		v<sup>2</sup> ∂<sup>2</sup>y/∂x<sup>2</sup> = ∂<sup>2</sup>y/∂t<sup>2</sup>
		+
		+	which is the general wave equation in one dimension that is used to describe situations involving periodic behavior.

	==Solutions to Wave Equation in One Dimension==		==Solutions to Wave Equation in One Dimension==

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 ∂<sup>2</sup>y/∂t<sup>2</sup> = -ω<sup>2</sup>Asin(ωt-kx)
-so 1/k<sup>2</sup> ∂<sup>2</sup>y/∂x<sup>2</sup> = 1/ω<sup>2</sup> ∂<sup>2</sup>y/∂t<sup>2</sup>
+so 1/k<sup>2</sup> ∂<sup>2</sup>y/∂x<sup>2</sup> = 1/ω<sup>2</sup> ∂<sup>2</sup>y/∂t<sup>2</sup> since -<sup>2</sup>Asin(ωt-kx) is common to both
 or 1/k<sup>2</sup> ∂<sup>2</sup>y/∂x<sup>2</sup> - 1/ω<sup>2</sup> ∂<sup>2</sup>y/∂t<sup>2</sup> = 0

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 k = 2π/λ where λ = wavelength
-w = 2πf = 2π/T where T = period
+ω = 2πf = 2π/T where T = period
 So if given the amplitude, wavelength and frequency of a particle that was behaving in a sine fashion, its position could be calculated at a particular time on the t axis using this equation. It would also give the particle position as a function of time at a particular distance on the x axis. Both plots y vs. x and y vs. t would be sine plots.

← Older revision		Revision as of 23:52, 5 May 2020
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	==Second Partial Derivatives of basic wave equation==		==Second Partial Derivatives of basic wave equation==
		+	Use a [https://en.wikibooks.org/wiki/Calculus/Tables_of_Derivatives table of derivatives] to find the second derivatives of the basic wave equation with respect to x and time.
		+
	∂<sup>2</sup>y/∂x<sup>2</sup> = -k<sup>2</sup>Asin(ωt-kx)		∂<sup>2</sup>y/∂x<sup>2</sup> = -k<sup>2</sup>Asin(ωt-kx)

← Older revision		Revision as of 23:50, 5 May 2020
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	==Partial Derivatives of basic wave equation==		==Partial Derivatives of basic wave equation==
		+	Use a [https://en.wikibooks.org/wiki/Calculus/Tables_of_Derivatives table of derivatives] to find the derivatives of the basic wave equation with respect to x and time.

	∂y/∂x = Akcos(kx-ωt)		∂y/∂x = Akcos(kx-ωt)

← Older revision		Revision as of 23:48, 5 May 2020
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	==Introduction==		==Introduction==
		+	Wave mechanics studies the properties of wave functions. These are periodic functions that carry information due to their periodic nature. The best way to study a function is to see how it changes, which can be done using derivatives, or derivatives of derivatives.

	==Basic Equation==		==Basic Equation==

← Older revision		Revision as of 19:31, 5 May 2020
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	==Solutions to Wave Equation in One Dimension==		==Solutions to Wave Equation in One Dimension==
		+	using fourier transform:
		+	# multiply both sides by e<sup>-ikx</sup>
		+	# integrate with respect to x
		+
		+	eventually end up with:

	y(x,t) = F(x-vt) + G(x+vt)		y(x,t) = F(x-vt) + G(x+vt)