- #1

Marioweee

- 18

- 5

- Homework Statement
- psb

- Relevant Equations
- $$H\Psi=E\Psi$$

A particle of mass m that is under the effect of a one-dimensional potential V (x) is described by the wave function:

\begin{array}{c} xe^{-bx}e^{-ict/\hbar }, x \geq 0\\ 0 , x \leq 0\end{array}

where $$b\geq 0,c\in R$$ and the wave function is normalized.

My solution:

First of all, I am new to quantum mechanics so i may have elementary errors. (Also I am not a native english speaker so i may have some grammar errors too).

As far as i know, the wave function of a stationary state is:

$$\Psi(x,t)=f(x)e^{-iEt/\hbar}$$

with E being the energy of the state.

In the problem the wave function given has the same form with c being E (i think here is my problem).

Then I have use the time independent Schröringer equation to obteein the value of E:

$$H\Psi=(-\dfrac{\hbar^2}{2m}\dfrac{\partial^2}{\partial x^2}+V(x))xe^{-bx}e^{-ict/\hbar }=e^{-ict/\hbar }(\dfrac{\hbar^2}{2m}be^{-bx}(2-bx)+V(x)xe^{-bx})=xe^{-bx}e^{-ict/\hbar }(\dfrac{\hbar^2}{m}\dfrac{b}{x}-\dfrac{\hbar^2}{2m}b^2+V(x))=Exe^{-bx}e^{-ict/\hbar }$$

so

$$E=(\dfrac{\hbar^2}{m}\dfrac{b}{x}-\dfrac{\hbar^2}{2m}b^2+V(x))$$

And if c=E, the form of the wave function would be:

$$\Psi(x,t)=f(x)e^{-i(\dfrac{\hbar^2}{m}\dfrac{b}{x}-\dfrac{\hbar^2}{2m}b^2+V(x))t/\hbar }$$

And now it has not the form of a stationary state because of the dependence on x and t.

So, would it be a stationary state? As I said before i think the problem is that c is not equal E necessarily.

I dindt try nº 2 and 3 yet because I want to understand stationary states first.

Thanks for the help

\begin{array}{c} xe^{-bx}e^{-ict/\hbar }, x \geq 0\\ 0 , x \leq 0\end{array}

where $$b\geq 0,c\in R$$ and the wave function is normalized.

- Is it a stationary state? What can you say about energy?
- It is possible to find stationary states with lower energy?
- Find V(x)

My solution:

First of all, I am new to quantum mechanics so i may have elementary errors. (Also I am not a native english speaker so i may have some grammar errors too).

As far as i know, the wave function of a stationary state is:

$$\Psi(x,t)=f(x)e^{-iEt/\hbar}$$

with E being the energy of the state.

In the problem the wave function given has the same form with c being E (i think here is my problem).

Then I have use the time independent Schröringer equation to obteein the value of E:

$$H\Psi=(-\dfrac{\hbar^2}{2m}\dfrac{\partial^2}{\partial x^2}+V(x))xe^{-bx}e^{-ict/\hbar }=e^{-ict/\hbar }(\dfrac{\hbar^2}{2m}be^{-bx}(2-bx)+V(x)xe^{-bx})=xe^{-bx}e^{-ict/\hbar }(\dfrac{\hbar^2}{m}\dfrac{b}{x}-\dfrac{\hbar^2}{2m}b^2+V(x))=Exe^{-bx}e^{-ict/\hbar }$$

so

$$E=(\dfrac{\hbar^2}{m}\dfrac{b}{x}-\dfrac{\hbar^2}{2m}b^2+V(x))$$

And if c=E, the form of the wave function would be:

$$\Psi(x,t)=f(x)e^{-i(\dfrac{\hbar^2}{m}\dfrac{b}{x}-\dfrac{\hbar^2}{2m}b^2+V(x))t/\hbar }$$

And now it has not the form of a stationary state because of the dependence on x and t.

So, would it be a stationary state? As I said before i think the problem is that c is not equal E necessarily.

I dindt try nº 2 and 3 yet because I want to understand stationary states first.

Thanks for the help