Quantum Mechanics Principles and Consequences in Bohr's Model

1. If the probability of finding a particle between [x,x+dx] dt is |(x,t)|2 dx dt= *(x,t)(x,t) dx dt, then .... (A) ... must hold when integrated over all space. (B) ... it is sufficient to demand that  must be bounded. (C) ... there is no constraint on . (D) ...  must be a real function. (E) ... /x must be > 0 everywhere.

2. If the probability of finding a particle between [x,x+dx] dt is |(x,t)|2 dx dt= *(x,t)(x,t) dx dt, then .... (A) ... must hold when integrated over all space. (B) ... it is sufficient to demand that  must be bounded. (C) ... there is no constraint on . (D) ...  must be a real function. (E) ... /x must be > 0 everywhere. • guarantees that the probability of finding a particle in all of space • is unity  the particle exists!

3. Recall that we could never explain the stability of Bohr’s n=1 state. Now assume that we would make the radius of the electron orbit smaller and smaller somehow. What would be the consequence? (A) Making the radius smaller results in the electron moving so fast that Coulomb attraction cannot hold it  electron escapes (B) We know less and less about the momentum of the electron, since the characteristic length scale x becomes smaller. But potential energy and kinetic energy are still balanced to make a stable orbit. (C) If we get below the “critical” radius for n=1 according to Bohr’s model, the electron will collapse into the nucleus.

4. Recall that we could never explain the stability of Bohr’s n=1 state. Now assume that we would make the radius of the electron orbit smaller and smaller somehow. What would be the consequence? (A) Making the radius smaller results in the electron moving so fast that Coulomb attraction cannot hold it  electron escapes (see Problem Set 4) (B) We know less and less about the momentum of the electron, since the characteristic length scale x becomes smaller. But potential energy and kinetic energy are still balanced to make a stable orbit. (C) If we get below the “critical” radius for n=1 according to Bohr’s model, the electron will collapse into the nucleus.