Why Can't Free Particles Have Negative Energy States in Quantum Mechanics?

[Yedi]

Why is there, for a free particle no physical state, i.e, proper wave function, for a negative energy?

Is it because k is imaginary? (k = sqrt(-2mE)/(reduced Planck's constant)

 

[kuruman]

Don't forget that, for a free particle, the potential energy is implicitly V = 0. Can a free particle exist in a state where its energy E is everywhere in space less than its potential energy? What does the Schrodinger equation have to say about that?

 

[Dickfore]

Because those states are not normalizable.

 

[kuruman]

Because those states are not normalizable.

Sure.

 

[paranormal]

)

This is a complex question and the answer lies in the fundamental principles of quantum mechanics. In quantum mechanics, a free particle is described by a wave function, which is a mathematical representation of the particle's state. This wave function is determined by the Schrödinger equation, which includes the particle's energy as a key parameter.

In quantum mechanics, energy is a quantized quantity and can only take on discrete values. These values can be either positive or negative, but it is important to note that negative energy does not necessarily mean a negative physical state. In fact, negative energy states can still have positive probabilities and can contribute to the overall behavior of the particle.

The reason why there is no proper wave function for a negative energy state is due to the mathematical properties of the Schrödinger equation. When solving the equation for negative energy values, the wave function becomes complex and cannot be interpreted as a physical state. This is where the imaginary term for k (wave number) comes into play. It is a mathematical representation of the complex nature of the wave function for negative energy states.

In summary, the absence of a proper wave function for negative energy states is a result of the mathematical properties of the Schrödinger equation and does not indicate a lack of physical state for these particles.