Question

The spherical symmetric potential in which a particle is moving is V(r) = Br", where ß
is a positive constant. what is the relationship between the expectation value of kinetic and
potential energy of particle in stationary state is,
(a) (T) = (U
(b) (7)= }(U)
(0) (T)=
(d) (T) = 5(U)
(U)​

Answer 1

Explanation:

An important problem in quantum mechanics is that of a particle in a spherically symmetric potential, i.e., a potential that depends only on the distance between the particle and a defined center point. In particular, if the particle in question is an electron and the potential is derived from Coulomb's law, then the problem can be used to describe a hydrogen-like (one-electron) atom (or ion).

An important problem in quantum mechanics is that of a particle in a spherically symmetric potential, i.e., a potential that depends only on the distance between the particle and a defined center point. In particular, if the particle in question is an electron and the potential is derived from Coulomb's law, then the problem can be used to describe a hydrogen-like (one-electron) atom (or ion).In the general case, the dynamics of a particle in a spherically symmetric potential are governed by a Hamiltonian of the following form:

An important problem in quantum mechanics is that of a particle in a spherically symmetric potential, i.e., a potential that depends only on the distance between the particle and a defined center point. In particular, if the particle in question is an electron and the potential is derived from Coulomb's law, then the problem can be used to describe a hydrogen-like (one-electron) atom (or ion).In the general case, the dynamics of a particle in a spherically symmetric potential are governed by a Hamiltonian of the following form:{\displaystyle {\hat {H}}={\frac {{\hat {p}}^{2}}{2m_{0}}}+V(r)}{\hat {H}}={\frac {{\hat {p}}^{2}}{2m_{0}}}+V(r)

An important problem in quantum mechanics is that of a particle in a spherically symmetric potential, i.e., a potential that depends only on the distance between the particle and a defined center point. In particular, if the particle in question is an electron and the potential is derived from Coulomb's law, then the problem can be used to describe a hydrogen-like (one-electron) atom (or ion).In the general case, the dynamics of a particle in a spherically symmetric potential are governed by a Hamiltonian of the following form:{\displaystyle {\hat {H}}={\frac {{\hat {p}}^{2}}{2m_{0}}}+V(r)}{\hat {H}}={\frac {{\hat {p}}^{2}}{2m_{0}}}+V(r)where {\displaystyle m_{0}}m_{0} is the mass of the particle, {\displaystyle {\hat {p}}}{\hat {p}} is the momentum operator, and the potential {\displaystyle V(r)}V(r) depends only on {\displaystyle r}r, the modulus of the radius vector r. The quantum mechanical wavefunctions and energies (eigenvalues) are found by solving the Schrödinger equation with this Hamiltonian. Due to the spherical symmetry of the system, it is natural to use spherical coordinates {\displaystyle r}r, {\displaystyle \theta }\theta and {\displaystyle \phi }\phi . When this is done, the time-independent Schrödinger equation for the system is separable, allowing the angular problems to be dealt with easily, and leaving an ordinary differential equation in {\displaystyle r}r to determine the energies for the particular potential {\displaystyle V(r)}V(r) under discussion.

Answer 2

The potential energy of particle in stationary state is $(T)=\frac{5}{2} (U)$.

Explanation:

$V(r)=\beta r^{5}$

kinetic(T) and potential(U)

virial the,

$V\alpha r^{n}$

$2(T)=r(U)$

Average value $n=5$

$2(T)=5(U)$

$(T)=\frac{5}{2} (U)$.