| Summary: | In this work, we investigate the Shannon entropy of four recently proposed hyperbolic potentials through studying position and momentum entropies. Our analysis reveals that the wave functions of the single-well potentials <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>U</mi><mrow><mn>0</mn><mo>,</mo><mn>3</mn></mrow></msub></semantics></math></inline-formula> exhibit greater localization compared to the double-well potentials <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>U</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></semantics></math></inline-formula>. This difference in localization arises from the depths of the single- and double-well potentials. Specifically, we observe that the position entropy density shows higher localization for the single-well potentials, while their momentum probability density becomes more delocalized. Conversely, the double-well potentials demonstrate the opposite behavior, with position entropy density being less localized and momentum probability density showing increased localization. Notably, our study also involves examining the Bialynicki–Birula and Mycielski (BBM) inequality, where we find that the Shannon entropies still satisfy this inequality for varying depths <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>u</mi><mo>¯</mo></mover></semantics></math></inline-formula>. An intriguing observation is that the sum of position and momentum entropies increases with the variable <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>u</mi><mo>¯</mo></mover></semantics></math></inline-formula> for potentials <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>U</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></msub></semantics></math></inline-formula>, while for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>U</mi><mn>0</mn></msub></semantics></math></inline-formula>, the sum decreases with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>u</mi><mo>¯</mo></mover></semantics></math></inline-formula>. Additionally, the sum of the cases <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>U</mi><mn>0</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>U</mi><mn>3</mn></msub></semantics></math></inline-formula> almost remains constant within the relative value <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0.01</mn></mrow></semantics></math></inline-formula> as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>u</mi><mo>¯</mo></mover></semantics></math></inline-formula> increases. Our study provides valuable insights into the Shannon entropy behavior for these hyperbolic potentials, shedding light on their localization characteristics and their relation to the potential depths. Finally, we extend our analysis to the Fisher entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover><mi>F</mi><mo>¯</mo></mover><mi>x</mi></msub></semantics></math></inline-formula> and find that it increases with the depth <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>u</mi><mo>¯</mo></mover></semantics></math></inline-formula> of the potential wells but <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover><mi>F</mi><mo>¯</mo></mover><mi>p</mi></msub></semantics></math></inline-formula> decreases with the depth.
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