Hello I try to understand how my professor did this just like that. I also added my attempt.
Thanks for your help!
Professor:
[MATH]\omega^{\prime}=\omega \sqrt{\frac{m_{e f f}}{m_{eff}+m}} \Rightarrow \frac{\Delta w}{w}=-\frac{m}{2 m_{eff}}[/MATH]
My attempt:
[MATH]=\frac{\omega \sqrt{\frac{m_{eff}}{m_{eff}+m}}-\omega}{\omega}=\frac{\not \omega\left[\sqrt{\frac{m_{eff}}{m_{eff}+m}}-1\right]}{\not\omega }[/MATH]
[MATH]=\frac{\sqrt{m_{eff}}}{\sqrt{m_{eff} +m}}-1=\frac{\sqrt{m_{e f f}}-\sqrt{m_{eff}+m}}{\sqrt{m_{eff}+m}}[/MATH]
[MATH]=\frac{\sqrt{m_{eff}}-\sqrt{m_{eff}+m}}{\sqrt{m_{\text {eff}}+m}} \frac{\sqrt{m_{eff}}+\sqrt{m_{eff}+m}}{\sqrt{m_{eff}}+\sqrt{m_{eff}+m}}[/MATH]
[MATH]=-\frac{m}{\sqrt{m_{eff}+m}\left(\sqrt{m_{eff}}+\sqrt{m_{eff}+m}\right)}[/MATH]
Thanks for your help!
Professor:
[MATH]\omega^{\prime}=\omega \sqrt{\frac{m_{e f f}}{m_{eff}+m}} \Rightarrow \frac{\Delta w}{w}=-\frac{m}{2 m_{eff}}[/MATH]
My attempt:
[MATH]=\frac{\omega \sqrt{\frac{m_{eff}}{m_{eff}+m}}-\omega}{\omega}=\frac{\not \omega\left[\sqrt{\frac{m_{eff}}{m_{eff}+m}}-1\right]}{\not\omega }[/MATH]
[MATH]=\frac{\sqrt{m_{eff}}}{\sqrt{m_{eff} +m}}-1=\frac{\sqrt{m_{e f f}}-\sqrt{m_{eff}+m}}{\sqrt{m_{eff}+m}}[/MATH]
[MATH]=\frac{\sqrt{m_{eff}}-\sqrt{m_{eff}+m}}{\sqrt{m_{\text {eff}}+m}} \frac{\sqrt{m_{eff}}+\sqrt{m_{eff}+m}}{\sqrt{m_{eff}}+\sqrt{m_{eff}+m}}[/MATH]
[MATH]=-\frac{m}{\sqrt{m_{eff}+m}\left(\sqrt{m_{eff}}+\sqrt{m_{eff}+m}\right)}[/MATH]
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