- #1
davidbenari
- 466
- 18
I'm having a hard time understanding the De Broglie hypothesis in a mathematical form. So my book says that Individual matter waves have a frequency ##f=E/h## and a wavelength ##\lambda=h/p##.
This is said as if the individual component sine waves of my complete wave packet have these frequency and wavelength, but that makes no sense, because to get a wave packet you have to add an infinite number of sine wave with different wavelengths and frequencies!
What then do the equations mean? That ##\lambda=h/p## and ##f=E/h## is the wavelength and frequency of the "small frequency wave" within my wave packet? It surely can't be the wavelength of the envelope since I'm told that the phase velocity is ##f\lambda##. I don't like this though because it implies that all wave packets can have a well-defined wavelength within the envelope, which is a bold mathematical statement of which I have no proof.
Also I'm told the group velocity is ##\frac{d}{dk}\omega |_{k_o}##. Why is this evaluated at ##k_o##? This is used to say that ##v_g=\frac{d(v_p k)}{dk}=v_p+k\frac{dv_p}{dk}## evaluated at ##k_o##
Another quick question is related to slit experiments. The typical equations of interference when applied to QM are approximately true because wavepackets don't have constant amplitudes and therefore you're not exactly cancelling stuff on some fringes right?
I'm specifically worried about how to mathematically interpret the wavelength and frequency relations. I'm not specifically worried about physical interpretations about this.
Thanks a whole lot.
This is said as if the individual component sine waves of my complete wave packet have these frequency and wavelength, but that makes no sense, because to get a wave packet you have to add an infinite number of sine wave with different wavelengths and frequencies!
What then do the equations mean? That ##\lambda=h/p## and ##f=E/h## is the wavelength and frequency of the "small frequency wave" within my wave packet? It surely can't be the wavelength of the envelope since I'm told that the phase velocity is ##f\lambda##. I don't like this though because it implies that all wave packets can have a well-defined wavelength within the envelope, which is a bold mathematical statement of which I have no proof.
Also I'm told the group velocity is ##\frac{d}{dk}\omega |_{k_o}##. Why is this evaluated at ##k_o##? This is used to say that ##v_g=\frac{d(v_p k)}{dk}=v_p+k\frac{dv_p}{dk}## evaluated at ##k_o##
Another quick question is related to slit experiments. The typical equations of interference when applied to QM are approximately true because wavepackets don't have constant amplitudes and therefore you're not exactly cancelling stuff on some fringes right?
I'm specifically worried about how to mathematically interpret the wavelength and frequency relations. I'm not specifically worried about physical interpretations about this.
Thanks a whole lot.