Wave equations and Schrodinger's equation

I'm having some difficulty with quantum which stems from a weak math background as a shaky foundation upon which to start with an inherently difficult subject. So, if anyone would be willing to help me with some conceptual obstacles and thought exercises, I would be very grateful. These are not course questions or assigned problem work, they are questions I find I am asking myself as I read through my textbook and flounder in confusion.

What is a wave? What does this motion really mean, in physical terms? What does it mean for energy to travel in such a manner? When you have something that is traveling as a wave, and you characterize it with a wave function, are you looking at one slice of the motion because the energy is actually traveling with this wave characteristic in 360 degrees? Is this wave pattern how it behaves in every direction, and the only thing that is unique to a given energy is wavelength and amplitude? Why do wavelength and amplitude change and what affects this change? What changes amplitude and what changes wavelength?

Why do sine and cosine functions describe waves? Why do they describe every kind of wave motion possible? How do they describe waves? How does the sine or cosine function work such that it captures the behavior of something that oscillates and, what causes things to oscillate in this manner?

From all of this, if you have Acos( ((2pi)x/lambda) - ((2pi)frequency*time)), what do each of these components mean? Is the component involving frequency a decay of the oscillation? Do you have to even consider decay in quantum? Is this a nonsensical question? If you do consider a decay, is the decay representative of factors which change wave functions of particles as they change energy states?

Is the wave function always of the form Acos(((2pi)x/lambda) - ((2pi)frequency*time))? Do you ever use sine instead of cosine?

When you say that the schrodinger wave equation is a function of position and time but, you can hold one or the other constant to take the equation to be a function of either position or time, what does that mean? The text represents a cosine wave if position is on the x axis and psi, the wave equation, is on the y axis, while time is constant, but, it shows only arrows propagating along the y axis, marked to indicate psi, the wave equation, when position is held constant and time is varied. (Quantum Mechanics, An Accessible Introduction. Robert Scherrer) I don't understand what this means, how it works that way, or why it works.

If anyone can offer me a dumbed-down explanation of the wave equation, or wave equations in general, I would really appreciate it.

Sincere thanks.
M.

What is a wave? What does this motion really mean, in physical terms? What does it mean for energy to travel in such a manner?
The energy in a wave is related to the amplitude. If an object follows a wave as the wave goes by the object moves back and forth, like a boat on the ocean.

When you have something that is traveling as a wave, and you characterize it with a wave function, are you looking at one slice of the motion because the energy is actually traveling with this wave characteristic in 360 degrees? Is this wave pattern how it behaves in every direction, and the only thing that is unique to a given energy is wavelength and amplitude? Why do wavelength and amplitude change and what affects this change? What changes amplitude and what changes wavelength?

The energy is always measured as the movement perpendicular to the direction the wave is traveling. Again think of the boat on the ocean example.

Why do sine and cosine functions describe waves? Why do they describe every kind of wave motion possible? How do they describe waves? How does the sine or cosine function work such that it captures the behavior of something that oscillates and, what causes things to oscillate in this manner?
The sine or cosine of any number gives you a number from -1 to 1. In this way a variable will give you a wave of unit amplitude. Simply multiply this function by a factor A to give it an amplitude. 10*cos(x) gives you a wave with amplitude of 10.

http://ocw.mit.edu/OcwWeb/Physics/8-03Fall-2004/VideoLectures/ [Broken]

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But, only some things travel perpendicular to the direction of propagation, correct? Doesn't light travel in 360 degrees?

Well, I guess even taking one segment of infinitely many they are all planes perpendicular to the propagation of motion. I'm sorry. My momentary misunderstanding.

Thank you, lost conjugate. It is much appreciated.

I'm having some difficulty with quantum which stems from a weak math background as a shaky foundation upon which to start with an inherently difficult subject. So, if anyone would be willing to help me with some conceptual obstacles and thought exercises, I would be very grateful. These are not course questions or assigned problem work, they are questions I find I am asking myself as I read through my textbook and flounder in confusion.

What is a wave? What does this motion really mean, in physical terms? What does it mean for energy to travel in such a manner? When you have something that is traveling as a wave, and you characterize it with a wave function, are you looking at one slice of the motion because the energy is actually traveling with this wave characteristic in 360 degrees? Is this wave pattern how it behaves in every direction, and the only thing that is unique to a given energy is wavelength and amplitude? Why do wavelength and amplitude change and what affects this change? What changes amplitude and what changes wavelength?

Why do sine and cosine functions describe waves? Why do they describe every kind of wave motion possible? How do they describe waves? How does the sine or cosine function work such that it captures the behavior of something that oscillates and, what causes things to oscillate in this manner?

From all of this, if you have Acos( ((2pi)x/lambda) - ((2pi)frequency*time)), what do each of these components mean? Is the component involving frequency a decay of the oscillation? Do you have to even consider decay in quantum? Is this a nonsensical question? If you do consider a decay, is the decay representative of factors which change wave functions of particles as they change energy states?

Is the wave function always of the form Acos(((2pi)x/lambda) - ((2pi)frequency*time))? Do you ever use sine instead of cosine?

When you say that the schrodinger wave equation is a function of position and time but, you can hold one or the other constant to take the equation to be a function of either position or time, what does that mean? The text represents a cosine wave if position is on the x axis and psi, the wave equation, is on the y axis, while time is constant, but, it shows only arrows propagating along the y axis, marked to indicate psi, the wave equation, when position is held constant and time is varied. (Quantum Mechanics, An Accessible Introduction. Robert Scherrer) I don't understand what this means, how it works that way, or why it works.

If anyone can offer me a dumbed-down explanation of the wave equation, or wave equations in general, I would really appreciate it.

Sincere thanks.
M.

Wave motion is a way to transfer energy through a continuous elastic medium without the transfer of mass. The continuum vibrates with a given frequency as the energy passes through it. The frequency is given by the source, while the wave speed and the wavelength are characteristics of the supporting medium. The wave function gives the displacement of the vibrating medium and the amplitude is the maximum displacement.

Sine and cosine functions “describe waves” because they are solutions of the wave equation, which describes all kinds of mechanical waves as well as electromagnetic waves. For example, $$y = ACos({\textstyle{{2\pi x} \over \lambda }} - 2\pi ft)$$ might be the displacement of a vibrating string. But, a wavefunction can have different forms. The actual form of the wavefunction for the vibrating string depends on how we start it; we can pluck it in various ways or we could hit it with a hammer or we can connect one end to a tuning fork. (I remember doing this in freshman lab.)

Imagine a piece of tape at position x. Then $$y = ACos(const - 2\pi ft)$$ gives the displacement of the piece of tape as it vibrates in the y-direction. If we hold t constant, then $$y = ACos({\textstyle{{2\pi x} \over \lambda }} - const)$$ gives the shape of the entire string at time t. The component involving frequency has nothing to do with decay. A decaying wavefunction would look like $$y = A(t)Cos({\textstyle{{2\pi x} \over\lambda }} - 2\pi ft)$$ where A(t) is a decreasing function of time.

The Schrodinger equation is not really a wave equation. In one dimension, the wave equation is $${{d^2 y} \over {d t^2}} = v^2 {{d^2 y} \over {d x^2}}$$ while the Schrodinger equation, in its simplest form, is $$i\hbar {\textstyle{{d\psi } \over {dt}}} = - {\textstyle{{\hbar ^2 } \over {2m}}}{\textstyle{{d^2 \psi } \over {dx^2 }}}$$. The Schrodinger equation is first order in time while the wave equation is second order in time. Note, also, that $$\psi (x,t)$$ is necessarily complex. A real function like $$y = ACos({{2\pi x} \over \lambda } - 2\pi ft)$$ is not a solution of the Schrodinger equation. $$\psi (x,t)$$ is used to calculate probabilities, so some people call them probability waves. As far as we know, Schrodinger waves are strictly mathematical and there is no vibrating medium.

I am not familiar with Scherrer’s text, but I did find that his section 3.2 The Meaning of the Wave Function