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质量为$m$的粒子处于能量为$E$的本征态,波函数为$\psi \left( x \right) =Axe^{-\frac{1}{2}a^2x^2}$,问粒子在什么样的位势中运动？
设粒子的波函数$\psi \left( x \right) =A\left( \frac{x}{a} \right) ^ne^{-x/a}$是一维势$V(x)$中的粒子能量本征态，其中$A$,$a$和$n$为常数. 当$x\rightarrow \infty$时，$V\rightarrow 0$.试求势能$V(x)$和粒子能量$E$. 粒子质量为$\mu$.
The following diagram represents a sundial, for which the triangle, called a gnomon, casts a shadow onto the surrounding surface, which has markings and numbers representing important information. It is known that this sundial is located either between the Tropic of Cancer and the Arctic Circle or between the Tropic of Capricorn and the Antarctic Circle.
Given that Neptune will be at opposition on 21 September 2024, calculate in which year Neptune was last at opposition near the time of the nort hern-hemisphere spring equinox. Assume that the orbits of Earth and Neptune are circular.
$(5points)$
An emission line of wavelength $ \lambda =600nm $ was observed in the spectrum of a white dwarf.Assuming that it originates from the interaction of a free non-relativistic electron with a magnetic field
$(a).$ calculate the magnetic flux density of the field;
$(b).$ estimate the wavelength of another spectral line, the discovery of which could confirm that the lines originate from particles of a plasma interacting with the magnetic field.
$(5points)$
