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jasony
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Homework Statement
Why is the ground state always symmetric and first excited state anti-symmetric?
OR Why does the ground state always have no node and first excited state has one node?
jasony said:Homework Statement
Why is the ground state always symmetric and first excited state anti-symmetric?
OR Why does the ground state always have no node and first excited state has one node?
Homework Equations
The Attempt at a Solution
jpr0 said:Hi, I know this is an old thread, but I have a simple Hamiltonian whose ground state is anti-symmetric.
[tex]H=\left[\begin{array}{cc}0 & \Delta\\\Delta & 0\end{array}\right]\,.[/tex]
Eigenstates and eigenenergies are:
[tex]\psi_{\eta}=\left[\begin{array}{c}1\\ \eta\end{array}\right]\,,\quad\varepsilon_{\eta}=\eta\Delta\,,[/tex]
where [itex]\eta=\pm 1[/itex]. The ground state corresponds to [itex]\eta=-1[/itex], which is anti-symmetric.
The ground state, also known as the lowest energy state, is always symmetric because of the fundamental principles of quantum mechanics. In quantum mechanics, particles are described by wave functions, and the ground state is the state in which the wave function has the lowest energy. Symmetry is a fundamental aspect of wave functions, and in order for the ground state to have the lowest energy, it must also be symmetric.
Symmetry plays a crucial role in determining the energy level of the ground state. In quantum mechanics, symmetrical states have lower energy levels compared to asymmetrical ones. This means that the ground state, being the state with the lowest energy, must also be symmetrical.
No, the ground state cannot be asymmetrical. As mentioned before, the ground state is always the state with the lowest energy, and in quantum mechanics, symmetrical states have lower energy levels. Any asymmetrical state would have a higher energy level, making it not the ground state.
If the ground state is not symmetric, it would not be the lowest energy state. This means that the system would not be in its most stable state, and it would be susceptible to changes and fluctuations. In order for a system to be in its most stable state, the ground state must be symmetric.
No, there are no exceptions to this rule. The ground state is always symmetric because of the fundamental principles of quantum mechanics. Symmetry is a fundamental aspect of wave functions, and in order for the ground state to have the lowest energy, it must also be symmetric.