How Can Quantum Potential Be Tailored for Specific Energy Levels?

[eljose]

let be the next problem: given a particle with mass so $$\hbar=(2m)^{0.5}$$ then we would have the quantum Hamiltonian:

$$H\phi(x)=E_{n}\phi(x)$$ with $$H=-D^{2}\phi+V(x)\phi$$

my question is how i would choose the potential so we have that the energies are the root of the equation $$f(x)=0$$

i try using the WKB approach to calculate the function:

$$\phi(x)=e^{iS(x)/\hbar}$$ with $$s^{2}=E_{n}-V(x)$$

with that i get a functional equation for the potential V(x), my problem is how i introduce the condition that the energies are the roots of f(x)...

 

[anathema]

Thank you for bringing up this interesting problem. The choice of potential in quantum mechanics is a crucial factor in determining the energy levels of a particle. In order to have the energies as the roots of the equation f(x)=0, we need to consider the potential V(x) as a function of x that satisfies the condition f(x)=0. This means that the potential should have a specific form or structure that allows for the energies to be the roots of this equation.

One approach to finding such a potential is through the WKB (Wentzel-Kramers-Brillouin) approximation, which you have already mentioned in your post. This method is based on the idea that the wave function can be written as a product of two functions, one representing the amplitude and the other representing the phase. By substituting this form of the wave function into the Schrödinger equation, we can obtain a functional equation for the potential V(x).

Now, in order to introduce the condition that the energies are the roots of f(x)=0, we need to solve this functional equation for V(x). This can be a challenging task and may require numerical methods or approximations. However, once we have the potential V(x) that satisfies the condition, we can then use it in the Schrödinger equation to obtain the energy levels E_n as the roots of f(x)=0.

I hope this helps in your research and leads you to a potential that satisfies your desired condition. Good luck!

 

[thepatientmental]

The potential in quantum mechanics refers to the energy function that determines the behavior of a particle in a given system. It is a crucial concept as it allows us to predict the energy levels and corresponding wavefunctions of a particle. In the given problem, the potential is represented by V(x) and is responsible for determining the energies (E_n) of the particle.

The potential in this case can be chosen in a way that the energies satisfy the equation f(x) = 0. This means that the energy levels are the roots of the function f(x). To achieve this, we can use the WKB (Wentzel-Kramers-Brillouin) approach, which is a semiclassical method for solving the Schrödinger equation.

Using the WKB approach, we can write the wavefunction as \phi(x) = e^{iS(x)/\hbar}, where S(x) is the classical action function. Substituting this into the Schrödinger equation, we get a functional equation for the potential V(x). Solving this equation for V(x) will give us the required potential that satisfies the condition of having energy levels as the roots of f(x).

In summary, the potential in quantum mechanics plays a crucial role in determining the energy levels and wavefunctions of a particle. To choose the potential in a way that satisfies the condition of having energy levels as the roots of a given function, we can use the WKB approach to solve the functional equation for the potential.