Question

Three commonly discussed discrete symmetry transformations are: Time reversal \(t \rightarrow \tilde{t}=(-1)^{\mathrm{n}} t\) for integer \(\mathfrak{n}\) Parity \(y \rightarrow \tilde{y}=(-1)^{\mathrm{n}} y\) for integer \(\mathfrak{n}\) Charge conjugation \(y \rightarrow \tilde{y}=y^{*}\) Verify that the wave equation, \(\ddot{y}+\omega_{0}^{2} y=0\), is invariant upon each of these discrete transformations.

Short answer

The wave equation is invariant under time reversal, parity, and charge conjugation transformations.

Step-by-step solution

Apply Time Reversal Transformation

For the wave equation, under time reversal the coordinates transform as \( t \rightarrow \tilde{t} = (-1)^{\mathfrak{n}} t \). The second time derivative transforms as \( \ddot{y}(t) = \frac{d^2 y}{dt^2} \rightarrow \ddot{y}(\tilde{t}) = \frac{d^2 y}{d((-1)^{\mathfrak{n}} t)^2} = (-1)^{2\mathfrak{n}}\frac{d^2 y}{dt^2} = \ddot{y}(t) \). Since \( (-1)^{2\mathfrak{n}} = 1 \), the time-reversed form of the wave equation remains \( \ddot{y}(t) + \omega_0^2 y(t) = 0 \). Therefore, the equation is invariant under time reversal.

Apply Parity Transformation

Under parity transformation, the spatial coordinate transforms as \( y \rightarrow \tilde{y} = (-1)^{\mathfrak{n}} y \). For \(\ddot{y}\), no spatial derivative is involved, so \( \ddot{y}\) remains unchanged. The transformed wave equation becomes \( \ddot{\tilde{y}} + \omega_0^2 \tilde{y} = 0 \), which substitutes back to \( \ddot{y} + \omega_0^2 (-1)^{\mathfrak{n}} y = 0 \). Since multiplication by \((-1)^{\mathfrak{n}}\) does not change the form when squared, the equation stays the same, proving invariance.

Apply Charge Conjugation Transformation

For Charge Conjugation transformation, \( y \rightarrow \tilde{y} = y^* \). Since the wave equation involves real variables (assuming no complex terms in given wave context), conjugation does not alter the second derivative, i.e., \( \ddot{(y^*)} = (\ddot{y})^* = \ddot{y} \), and \( y^* \) simply equals \( y \) if \( y \) is real. Thus, the equation \( \ddot{y} + \omega_0^2 y = 0 \) maintains its form, showing invariance under charge conjugation.

Key concepts

Time Reversal

Time reversal is a fascinating symmetry transformation found in many physical equations. In the context of the given wave equation, we apply time reversal by transforming the time variable as follows: \( t \rightarrow \tilde{t} = (-1)^{\mathfrak{n}} t \). This essentially means flipping the direction of time, which is like watching a movie backwards. Now, when time is reversed, the second time derivative remains unchanged because \((-1)^{2\mathfrak{n}} = 1\).

This means even though we are conceptually reversing time, the mathematical form of the wave equation does not change. This is a clear indication that the wave equation is invariant under time reversal transformation. It's intriguing because, despite the reversal, the physics, as described by the equation, continues unchanged. This shows how symmetry can provide robust insights into the nature of equations and the systems they describe.

Parity Transformation

In the realm of symmetry transformations, parity transformation is key. It involves flipping the spatial coordinate, which means imagining the mirror image of the system. Under parity, the transformation is given by \( y \rightarrow \tilde{y} = (-1)^{\mathfrak{n}} y \). This would be like looking at a system's reflection in a mirror.

Applying this to the wave equation, we find that the parity transformation does not affect the second time derivative \( \ddot{y} \). The transformed wave equation thus becomes \( \ddot{y} + \omega_0^2 (-1)^{\mathfrak{n}} y = 0 \), which mirrors the original statement of the equation upon evaluation. Again, due to the nature of \((-1)^{2\mathfrak{n}} = 1\), the form of the equation remains unchanged.

This tells us that the wave equation remains invariant with respect to parity transformations as well. Parity is especially important in physics as it relates to the conservation laws and the symmetry properties that can make complex systems much simpler to understand.

Charge Conjugation

Charge conjugation is a transformation that involves changing the sign of charge particles, effectively transforming particles into antiparticles. In our current context, it simplifies to the transformation \( y \rightarrow \tilde{y} = y^* \), where \( y^* \) represents the complex conjugate.

In the case of the given wave equation, if we assume \( y \) to be a real number, then its complex conjugate \( y^* \) is simply \( y \) itself. This ensures that any involved second derivatives remain unchanged because real numbers don't change upon conjugation.

Thus, the wave equation \( \ddot{y} + \omega_0^2 y = 0 \) stays intact. This conservation of form indicates that the wave equation respects charge conjugation symmetry. Such transformations bring forth the idea that underlying structures in physical laws can be preserved even when the charge attributes of particles change, highlighting another layer of consistent symmetry in physics.